/* Copyright 2014-2018 The PySCF Developers. All Rights Reserved.
  
   Licensed under the Apache License, Version 2.0 (the "License");
    you may not use this file except in compliance with the License.
    You may obtain a copy of the License at
 
        http://www.apache.org/licenses/LICENSE-2.0
 
    Unless required by applicable law or agreed to in writing, software
    distributed under the License is distributed on an "AS IS" BASIS,
    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    See the License for the specific language governing permissions and
    limitations under the License.

 *
 * Author: Qiming Sun <osirpt.sun@gmail.com>
 */

#include <stdlib.h>
#include <stdint.h>
#include <math.h>
#include <complex.h>
#include "cint.h"
#include "vhf/fblas.h"

//#define ATOM_OF         0
//#define ANG_OF          1
#define RADI_POWER      3
#define SO_TYPE_OF      4

#define ECP_LMAX        5
//#define PTR_EXP         5
//#define PTR_COEFF       6
#define CART_MAX        128 // ~ lmax = 14
#define SIM_ZERO        1e-50
#define EXPCUTOFF       39   // 1e-17
#define CUTOFF          460  // ~ 1e200
#define CLOSE_ENOUGH(x, y)      (fabs(x-y) < 1e-12*fabs(y) || fabs(x-y) < 1e-12)
#define SQUARE(r)       (r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
#define CART_CUM        (455+1) // upto l = 12
#define K_TAYLOR_MAX    7
#define K_TAB_COL       24      // >= (7*2+1+K_TAYLOR_MAX)
#define K_TAB_ENTRIES   400
#define K_TAB_INTERVAL  (16./K_TAB_ENTRIES)    // [0,16], interval 0.04
#define MAX(X,Y)        ((X)>(Y)?(X):(Y))
#define MALLOC_INSTACK(var, n) \
                var = (void *)cache; \
                cache = (void *)(((uintptr_t)(var + (n)) + 7) & (-(uintptr_t)8));
#define MARK_STACK      cache0 = cache;
#define RESTORE_STACK   cache = cache0;

// Held in env, to get *ecpbas, necpbas
#define AS_ECPBAS_OFFSET        18
#define AS_NECPBAS              19


// for radial grids
#define LEVEL0          5
//#define LEVEL_MAX       11      // 2047 points
#define LEVEL_MAX       11

double CINTcommon_fac_sp(int);

typedef struct {
    double *u_ecp;
} ECPOpt;

typedef int Function_cart(double *gctr, int *shls, int *ecpbas, int necpbas,
                          int *atm, int natm, int *bas, int nbas, double *env,
                          ECPOpt *opt, double *cache);

static double _sph_ine_tab[] = { // 400x24
9.802640211919197e-01, 6.534919212092035e-03, 2.613937811451498e-05, 7.468346329123935e-08, 1.659625812043017e-10, 3.017493035919202e-13, 4.642287455711235e-16, 6.189706898282851e-19, 7.281999097680386e-22, 7.665254523625293e-25, 7.300236357712470e-28, 6.348027199385843e-31, 5.078418750076879e-34, 3.761789744992798e-37, 2.594336600846831e-40, 1.673764894480139e-43, 1.014402615043443e-46, 5.796584581226479e-50, 3.133288094277013e-53, 1.606814005366913e-56, 7.838115320992611e-60, 3.645634279400473e-63, 1.620281595519583e-66, 6.894814102547425e-70,
9.423296940236874e-01, 1.884207224818979e-02, 2.260816141980705e-04, 1.937731683913445e-06, 1.291774150186003e-08, 7.045863444090539e-11, 3.251876940391953e-13, 1.300732413080320e-15, 4.590769115454675e-18, 1.449703482841226e-20, 4.141979079274455e-23, 1.080509516656789e-25, 2.593209009591830e-28, 5.762660193098549e-31, 1.192269748342981e-33, 2.307610747161589e-36, 4.195642826638227e-39, 7.192510565377163e-42, 1.166350154849031e-44, 1.794380813740386e-47, 2.625917780013486e-50, 3.664064504096460e-53, 4.885411023215048e-56, 6.236685174215352e-59,
9.063462346100909e-01, 3.019141928900223e-02, 6.036559400239012e-04, 8.622287807164008e-06, 9.579352242057132e-08, 8.707893125883569e-10, 6.698035852053903e-12, 4.465182134941749e-14, 2.626496412795802e-16, 1.382331888858186e-18, 6.582396524955082e-21, 2.861861761817600e-23, 1.144727746014053e-25, 4.239678246653739e-28, 1.461941754380445e-30, 4.715895044881049e-33, 1.429046731905972e-35, 4.082959134165019e-38, 1.103494821514312e-40, 2.829456206258353e-43, 6.901073554470799e-46, 1.604892532642074e-48, 3.566410987925639e-51, 7.588075536423757e-54,
8.722009233724092e-01, 4.064962396752648e-02, 1.137552639698620e-03, 2.274397829003244e-05, 3.537251969982054e-07, 4.501340147805141e-09, 4.847109923004117e-11, 4.523621584594117e-13, 3.725109389920223e-15, 2.744682624160316e-17, 1.829714169367008e-19, 1.113701098036058e-21, 6.236545062744792e-24, 3.233683163047443e-26, 1.561054390080907e-28, 7.049787985018583e-31, 2.990768393494177e-33, 1.196289251644110e-35, 4.526438390070580e-38, 1.624855402807172e-40, 5.548225060005377e-43, 1.806380559727397e-45, 5.619798551249213e-48, 1.673968300753623e-50,
8.397879831360249e-01, 5.027877721939270e-02, 1.808362812813033e-03, 4.647686347510318e-05, 9.292332256868268e-07, 1.520219076183037e-08, 2.104569083041292e-10, 2.525162087703369e-12, 2.673432884835239e-14, 2.532520256428454e-16, 2.170586052040690e-18, 1.698623809823192e-20, 1.222950443880237e-22, 8.152665619678108e-25, 5.060092852017936e-27, 2.938025380983860e-29, 1.602514346193933e-31, 8.241296164981470e-34, 4.009189198003429e-36, 1.850357521979083e-38, 8.123371541285905e-41, 3.400424173842929e-43, 1.360148833507746e-45, 5.209007356442930e-48,
8.090081339019514e-01, 5.913671199826918e-02, 2.598424834644346e-03, 8.160211998856853e-05, 1.993744098983762e-06, 3.986139377821739e-08, 6.744100728921062e-10, 9.889470956573337e-12, 1.279622166510847e-13, 1.481488080423163e-15, 1.551879635695967e-17, 1.484281679217320e-19, 1.306074233139604e-21, 1.064142859516303e-23, 8.072373323154131e-26, 5.728510053119174e-28, 3.818846680199104e-30, 2.400328205168628e-32, 1.427174308308297e-34, 8.050483194653290e-37, 4.319652884952700e-39, 2.209999687114206e-41, 1.080419567502579e-43, 5.057176802023883e-46,
7.797681769803957e-01, 6.727729617192950e-02, 3.491682688901381e-03, 1.295521545952391e-04, 3.740065183404311e-06, 8.835978508986519e-08, 1.766583448475031e-09, 3.061266611360877e-11, 4.680957668323452e-13, 6.404436109277377e-15, 7.928192361364623e-17, 8.961250982568824e-19, 9.318767845088264e-21, 8.972853685685055e-23, 8.044022614751631e-25, 6.746153851558893e-27, 5.314840467927241e-29, 3.947961127780111e-31, 2.774113001703102e-33, 1.849330487722149e-35, 1.172701198775740e-37, 7.090503732787043e-40, 4.096604557506365e-42, 2.266140260236755e-44,
7.519806065099561e-01, 7.475070512352422e-02, 4.473555274713588e-03, 1.914505449644170e-04, 6.375892210525950e-06, 1.737786486384450e-07, 4.008427116300951e-09, 8.014026540370688e-11, 1.413846115606957e-12, 2.231885264598680e-14, 3.187813611266366e-16, 4.157367122234428e-18, 4.988175533040940e-20, 5.541780336446298e-22, 5.732302392725156e-24, 5.546901453139527e-26, 5.042244809781671e-28, 4.321623796889308e-30, 3.503800774801065e-32, 2.695079679934517e-34, 1.971908861923835e-36, 1.375686386093325e-38, 9.170852340169328e-41, 5.853506792622323e-43,
7.255632465211918e-01, 8.160367547530310e-02, 5.530815856752632e-03, 2.681481701173305e-04, 1.011823668994520e-05, 3.124930305455487e-07, 8.168054648039442e-09, 1.850587087464627e-10, 3.699850401381123e-12, 6.618867740655736e-14, 1.071369838211224e-15, 1.583445821705979e-17, 2.153117630614553e-19, 2.710933129251171e-21, 3.177926797994465e-23, 3.485074323629212e-25, 3.590323325355264e-27, 3.487431373375834e-29, 3.204409977879080e-31, 2.793386250709058e-33, 2.316314775123981e-35, 1.831395301402106e-37, 1.383645271997669e-39, 1.000884640136975e-41,
7.004389118290668e-01, 8.787974125233637e-02, 6.651480889569160e-03, 3.602558632684483e-04, 1.518867146616316e-05, 5.241706487945403e-07, 1.531058000541370e-08, 3.876486093346255e-10, 8.661215889008114e-12, 1.731616684729988e-13, 3.132465358171466e-15, 5.174078457568593e-17, 7.862917483701786e-19, 1.106428817416875e-20, 1.449570476616633e-22, 1.776642094452763e-24, 2.045574577993796e-26, 2.220661949927553e-28, 2.280451657869560e-30, 2.221777914010818e-32, 2.059040163516108e-34, 1.819481118591509e-36, 1.536345836110685e-38, 1.242074078030305e-40,
6.765350911558576e-01, 9.361945371114844e-02, 7.824707504797336e-03, 4.681738921325179e-04, 2.180930258870306e-05, 8.316938031664069e-07, 2.684583910669537e-08, 7.511641496452160e-10, 1.854804793764511e-11, 4.098283595804409e-13, 8.193575672782376e-15, 1.495759413219887e-16, 2.512219435375146e-18, 3.907016870618316e-20, 5.657328347997245e-22, 7.663446202166263e-24, 9.751987792903223e-26, 1.170079170870446e-27, 1.328035649504993e-29, 1.430034493807493e-31, 1.464766837491307e-33, 1.430572088364921e-35, 1.335089272208570e-37, 1.192967128812958e-39,
6.537836509950933e-01, 9.886058597209520e-02, 9.040699003168024e-03, 5.921185463558279e-04, 3.019938470977199e-05, 1.261019425506075e-06, 4.457236071368571e-08, 1.365753162782863e-09, 3.693149250538184e-11, 8.936571491864001e-13, 1.956677811748755e-14, 3.911916489249546e-16, 7.195671239819596e-18, 1.225598043243590e-19, 1.943594694173485e-21, 2.883447429149476e-23, 4.018614857945507e-25, 5.280745363948835e-27, 6.564288540009260e-29, 7.741469848656017e-31, 8.684509313319550e-33, 9.289389581578651e-35, 9.494870612442851e-37, 9.291998410785926e-39,
6.321205588285578e-01, 1.036383235143269e-01, 1.029061774259589e-02, 7.321460883680680e-04, 4.057250544293857e-05, 1.840990395173499e-06, 7.071674912159513e-08, 2.354918012025038e-09, 6.920876084399665e-11, 1.820143329151406e-12, 4.331433624321587e-14, 9.412069363388704e-16, 1.881717162782282e-17, 3.483549477293463e-19, 6.004450438117013e-21, 9.682231855957900e-23, 1.466687423114293e-24, 2.094863403563336e-26, 2.830406199582482e-28, 3.628158722983189e-30, 4.423956555945650e-32, 5.143471077554361e-34, 5.714292488971905e-36, 6.078374796471110e-38,
6.114856244028342e-01, 1.079854415178434e-01, 1.156650485925946e-02, 8.881743024780179e-04, 5.313427158145224e-05, 2.603109453813832e-06, 1.079678926519672e-07, 3.882408488694344e-09, 1.232124104576527e-10, 3.499270583056130e-12, 8.992697975180006e-14, 2.110259375015334e-15, 4.556192707286104e-17, 9.108994199153757e-19, 1.695607709223078e-20, 2.952797770562511e-22, 4.830631594144867e-24, 7.451296962019681e-26, 1.087266909887928e-27, 1.505172046838923e-29, 1.982093931505646e-31, 2.488766216723909e-33, 2.986107810512032e-35, 3.430412972161793e-37,
5.918222578598231e-01, 1.119324699680684e-01, 1.286120630084882e-02, 1.060001857302613e-03, 6.808043685177060e-05, 3.581285464793354e-06, 1.595056229311132e-07, 6.159433578748354e-09, 2.099269290005269e-10, 6.402901146704448e-12, 1.767190222777191e-13, 4.453788373237586e-15, 1.032764424354891e-16, 2.217578604434540e-18, 4.433498767431274e-20, 8.292207189026603e-22, 1.456993986020573e-23, 2.413820288857954e-25, 3.782944683593620e-27, 5.624734620245826e-29, 7.955423938391327e-31, 1.072870693336584e-32, 1.382591240785306e-34, 1.705927272952057e-36,
5.730772435975399e-01, 1.155078473741702e-01, 1.416830469026474e-02, 1.247325678486786e-03, 8.559541702683246e-05, 4.811560355346477e-06, 2.290236255239660e-07, 9.452078231060181e-09, 3.443134821874354e-10, 1.122468721114226e-11, 3.311321685597624e-13, 8.920211537400009e-15, 2.210953981490762e-16, 5.074515259838354e-18, 1.084432851478003e-19, 2.168051312210924e-21, 4.071953725407970e-23, 7.211035260361866e-25, 1.208013913366905e-26, 1.919970914203807e-28, 2.902737870906804e-30, 4.184511847918588e-32, 5.764279955245770e-34, 7.602674982708987e-36,
5.552005288137498e-01, 1.187380638926901e-01, 1.548205657424927e-02, 1.449756512013810e-03, 1.058511437997683e-04, 6.331823835150759e-06, 3.207465472556373e-07, 1.408881344881271e-08, 5.462416008028229e-10, 1.895403419454607e-11, 5.951618689207405e-13, 1.706563797705505e-14, 4.502424476095938e-16, 1.099981002498531e-17, 2.502193147402336e-19, 5.324983368983020e-21, 1.064595304250064e-22, 2.006847732699146e-24, 3.578702429411037e-26, 6.054616656852117e-28, 9.744049075136297e-30, 1.495261691688425e-31, 2.192598984437593e-33, 3.078386627927686e-35,
5.381450257559952e-01, 1.216477945925827e-01, 1.679733464492681e-02, 1.666832843105426e-03, 1.290062138725492e-04, 8.181521886936468e-06, 4.394413635475253e-07, 2.046799248242602e-08, 8.415246383961493e-10, 3.096554994811516e-11, 1.031139804452122e-12, 3.135581455149412e-14, 8.773263316007978e-16, 2.273128003705574e-17, 5.483873143620322e-19, 1.237701348583612e-20, 2.624314178610683e-22, 5.246643814328488e-24, 9.922714464392029e-26, 1.780454578414938e-27, 3.038956080231845e-29, 4.945885648569614e-31, 7.691818247961134e-33, 1.145346308884449e-34,
5.218664267676943e-01, 1.242600235462328e-01, 1.810957455323678e-02, 1.898033321659904e-03, 1.552052942917376e-04, 1.040136405768962e-05, 5.904231639189556e-07, 2.906523208634183e-08, 1.263054060675248e-09, 4.912528704558056e-11, 1.729123018449930e-12, 5.557976524470974e-14, 1.643828411653500e-15, 4.502160776715399e-17, 1.148128257340755e-18, 2.739227677304166e-20, 6.139600917126357e-22, 1.297538585654849e-23, 2.594093083152433e-25, 4.920440786318391e-27, 8.878038765239024e-29, 1.527416388661771e-30, 2.511101393595806e-32, 3.952709345915435e-34,
5.063230312815611e-01, 1.265961593831042e-01, 1.941472596192939e-02, 2.142787832274750e-03, 1.845787492072705e-04, 1.303303372932079e-05, 7.795555886439546e-07, 4.044058525488137e-08, 1.852026050082128e-09, 7.591493257857583e-11, 2.816153937332171e-12, 9.540349655577745e-14, 2.973910687452203e-15, 8.584631692476067e-17, 2.307409287410072e-18, 5.802290566823627e-20, 1.370728800682073e-21, 3.053333168699244e-23, 6.434044708739933e-25, 1.286319681583461e-26, 2.446300822624709e-28, 4.436081525246771e-30, 7.686997322577973e-32, 1.275377605633448e-33,
4.914755839689684e-01, 1.286761428981434e-01, 2.070920751234611e-02, 2.400487335057321e-03, 2.172424569787300e-04, 1.611890480296700e-05, 1.013246207221353e-06, 5.524542018945295e-08, 2.659252536237624e-09, 1.145749259900188e-10, 4.467665737188296e-12, 1.590961351965341e-13, 5.213164602582841e-15, 1.581899958377264e-16, 4.469617682089592e-18, 1.181509833383892e-19, 2.934165638294525e-21, 6.870765092651597e-23, 1.522001187135905e-24, 3.198760453000486e-26, 6.395082698196713e-28, 1.219103902128570e-29, 2.220772400555977e-31, 3.873408478502617e-33,
4.772871233072715e-01, 1.305185472656688e-01, 2.198986540377530e-02, 2.670492592556639e-03, 2.532977899421938e-04, 1.970176758019172e-05, 1.298437172299717e-06, 7.422892914946797e-08, 3.746547599694751e-09, 1.692672950368288e-10, 6.921313997371905e-12, 2.584648684449999e-13, 8.881469191674576e-15, 2.826245009714687e-16, 8.374393733114801e-18, 2.321541571323740e-19, 6.046208575737749e-21, 1.484792261579577e-22, 3.449371634820574e-24, 7.602791567712239e-26, 1.594063416186110e-27, 3.186909150552418e-29, 6.088409099002073e-31, 1.113691772263508e-32,
4.637228398768964e-01, 1.321406713709966e-01, 2.325393530690784e-02, 2.952141888175205e-03, 2.928317322118046e-04, 2.382456605715973e-05, 1.642591513185447e-06, 9.824420003660245e-08, 5.188179242073002e-09, 2.452587974457146e-10, 1.049351821902704e-11, 4.100390017502337e-13, 1.474372985440144e-14, 4.909502390826970e-16, 1.522268192052498e-17, 4.415994213366805e-19, 1.203518559486136e-20, 3.092828584305877e-22, 7.518878116286068e-24, 1.734247610493291e-25, 3.805137481805247e-27, 7.960910042346302e-29, 1.591572684233594e-30, 3.046621178329281e-32,
4.507499437463384e-01, 1.335586267362804e-01, 2.449900735395414e-02, 3.244757832268982e-03, 3.359171136532075e-04, 2.853014835529238e-05, 2.053675452977424e-06, 1.282537928386357e-07, 7.072375765152941e-09, 3.491247454441267e-10, 1.559899553761352e-11, 6.365472633991269e-13, 2.390292252849138e-14, 8.312387052073418e-16, 2.691716615282967e-17, 8.154941094050264e-19, 2.321148096176872e-20, 6.229692578266206e-22, 1.581710652220588e-23, 3.810223780911893e-25, 8.731260975682391e-27, 1.907823433402614e-28, 3.983567564048736e-30, 7.964080601333788e-32,
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3.144654088050266e-02, 2.946877101380535e-02, 2.588639540619975e-02, 2.132839509990605e-02, 1.649653592825369e-02, 1.199073325372471e-02, 8.201060092343518e-03, 5.285463995833784e-03, 3.214773303821078e-03, 1.848284991748354e-03, 1.006130860851347e-03, 5.194329113786493e-04, 2.547499198633639e-04, 1.188827229142408e-04, 5.287359793352091e-05, 2.244660089712720e-05, 9.109784863650277e-06, 3.539500236720960e-06, 1.318432141308537e-06, 4.714505997136713e-07, 1.620438778599092e-07, 5.360160649000604e-08, 1.708355842152804e-08, 5.251912844171947e-09,
3.136762860727686e-02, 2.939977235838962e-02, 2.583442176441808e-02, 2.129612688648933e-02, 1.648229577913417e-02, 1.198993729977619e-02, 8.208185973767919e-03, 5.295682741496197e-03, 3.224795690051300e-03, 1.856440161140355e-03, 1.011974920812484e-03, 5.232235151515115e-04, 2.570099993266136e-04, 1.201338047898214e-04, 5.352112044799525e-05, 2.276162831605336e-05, 9.254465630074640e-06, 3.602446020415770e-06, 1.344452410843024e-06, 4.816970115580610e-07, 1.658963599795125e-07, 5.498742817286539e-08, 1.756139062987560e-08, 5.410102053392232e-09,
3.128911138923616e-02, 2.933109440618089e-02, 2.578264810897690e-02, 2.126393292026822e-02, 1.646803418895953e-02, 1.198907011046623e-02, 8.215232485885160e-03, 5.305839677643497e-03, 3.234782225268609e-03, 1.864581991187529e-03, 1.017819907836628e-03, 5.270214114272550e-04, 2.592784520902656e-04, 1.213917930131973e-04, 5.417342009049552e-05, 2.307957883144690e-05, 9.400770292319421e-06, 3.666223409260362e-06, 1.370869208207234e-06, 4.921207369407304e-07, 1.698236049100827e-07, 5.640310231536183e-08, 1.805055112031054e-08, 5.572388897716357e-09,
};

#define ORDER7OFFSET    8
static double _sph_ine_tab_order7[] = { // 400x8x8, expand ECPsph_ine_opt up to order 7
 9.8026402119191969e-01, 6.5349192120920351e-03, 2.6139378114514981e-05, 7.4683463291239355e-08, 1.6596258120430169e-10, 3.0174930359192018e-13, 4.6422874557112354e-16, 6.1897068982828512e-19,
-9.7372910197982765e-01, 3.2023718077062419e-01, 2.5878731168002735e-03, 1.1128001992975867e-05, 3.3026855408959629e-08, 7.5136040823133709e-11, 1.3880501389645580e-13, 2.1602149894998125e-16,
 1.2939662827504519e+00,-6.4308829935269984e-01, 1.2551367599264518e-01, 1.0979793491245177e-03, 4.9127935503807721e-06, 1.4937146675075184e-08, 3.4539483992972387e-11, 6.4559806614079017e-14,
-1.9370545821031517e+00, 1.1580861775979472e+00,-3.8209020812425043e-01, 5.2696403386895060e-02, 4.8308632669755771e-04, 2.2181696704892722e-06, 6.8595629751683660e-09, 1.6053920119069382e-11,
 3.0951407597010991e+00,-2.0584978437151644e+00, 8.7694252119556637e-01,-2.1617330039631805e-01, 2.2938769717294961e-02, 2.1737026586275069e-04, 1.0169196985922520e-06, 3.1850916175148835e-09,
-5.1536386034162636e+00, 3.6748397777459081e+00,-1.8300456389194231e+00, 6.0511367789001502e-01,-1.1889503085684590e-01, 1.0209897925470555e-02, 9.9309533441240569e-05, 4.7138144747566653e-07,
 8.8284783811621708e+00,-6.6127497381642781e+00, 3.6630497567517954e+00,-1.4573589693451083e+00, 3.9350660876655841e-01,-6.4198924933068918e-02, 4.6132048683245799e-03, 4.5873862878265983e-05,
-1.5441228119326450e+01, 1.1997609036386198e+01,-7.1825650336245719e+00, 3.2520983558196255e+00,-1.0768877756605337e+00, 2.4558185884604522e-01,-3.4218776603575786e-02, 2.1070725887895958e-03,
 9.4232969402368738e-01, 1.8842072248189790e-02, 2.2608161419807049e-04, 1.9377316839134449e-06, 1.2917741501860030e-08, 7.0458634440905395e-11, 3.2518769403919528e-13, 1.3007324130803201e-15,
-9.2348762177549759e-01, 2.9541854683583807e-01, 7.3119099240881941e-03, 9.4961770253260684e-05, 8.4833548392324935e-07, 5.8014194233285842e-09, 3.2194882442293417e-11, 1.5045597321540574e-13,
 1.2189061686113356e+00,-5.9837314747827841e-01, 1.1091248587239898e-01, 3.0391986746324926e-03, 4.1360118750538900e-05, 3.7982317956857146e-07, 2.6454643276949385e-09, 1.4874146513869068e-11,
-1.8172793160896141e+00, 1.0786168609303228e+00,-3.4843822565893084e-01, 4.4518358195681659e-02, 1.3096058606303291e-03, 1.8421673778491488e-05, 1.7266555078284553e-07, 1.2197133111721585e-09,
 2.8958961770199370e+00,-1.9166687833994813e+00, 8.0659598494846896e-01,-1.9310068012914899e-01, 1.8486565378438458e-02, 5.7694789862663064e-04, 8.3303021926115510e-06, 7.9361239482497811e-08,
-4.8125649604194187e+00, 3.4196981657051060e+00,-1.6891239063857508e+00, 5.4934842532331474e-01,-1.0398856326993432e-01, 7.8305801654959106e-03, 2.5799607630247782e-04, 3.8084155547410485e-06,
 8.2322631261245256e+00,-6.1499690901020792e+00, 3.3866122278617823e+00,-1.3326807070714561e+00, 3.5249374128334970e-01,-5.4957383792028344e-02, 3.3581685315328034e-03, 1.1660936998126413e-04,
-1.4382232216226605e+01, 1.1151798284051441e+01,-6.6462082881454876e+00, 2.9855109426027053e+00,-9.7532815764401248e-01, 2.1701353993802336e-01,-2.8660325236325203e-02, 1.4514722406969232e-03,
 9.0634623461009090e-01, 3.0191419289002229e-02, 6.0365594002390120e-04, 8.6222878071640075e-06, 9.5793522420571317e-08, 8.7078931258835689e-10, 6.6980358520539028e-12, 4.4651821349417492e-14,
-8.7615481532108863e-01, 2.7232642954104402e-01, 1.1478085148261291e-02, 2.5014213993017678e-04, 3.7368181637148698e-06, 4.2675374352681547e-08, 3.9522876709252972e-10, 3.0812383227510867e-12,
 1.1484812448621327e+00,-5.5672597788256606e-01, 9.7602571952114428e-02, 4.6711725339896419e-03, 1.0746117479100407e-04, 1.6560939157543097e-06, 1.9302756985549667e-08, 1.8136992002922571e-10,
-1.7052072227446988e+00, 1.0046214408046865e+00,-3.1749025958474708e-01, 3.7219907545368539e-02, 1.9695355591575888e-03, 4.7200423402148745e-05, 7.4514594178107065e-07, 8.8273456379934407e-09,
 2.7098286635493851e+00,-1.7846840214427508e+00, 7.4167078043384294e-01,-1.7216171276217007e-01, 1.4598868029562954e-02, 8.4844945581954490e-04, 2.1044418045323422e-05, 3.3895218792941198e-07,
-4.4945126849921362e+00, 3.1824074295817741e+00,-1.5588414166682454e+00, 4.9836282896499584e-01,-9.0643824003961010e-02, 5.7988784220065195e-03, 3.7073015120335159e-04, 9.4839471515876702e-06,
 7.6769201145739103e+00,-5.7198059356913165e+00, 3.1308220858799527e+00,-1.2182341926822216e+00, 3.1536001377840717e-01,-4.6798400159514246e-02, 2.3107820151120509e-03, 1.6360636721506695e-04,
-1.3396726050265226e+01, 1.0365994031135923e+01,-6.1496849757658119e+00, 2.7402208087898625e+00,-8.8279654394801343e-01, 1.9140428752157862e-01,-2.3821947891002823e-02, 9.1705648089344472e-04,
 8.7220092337240918e-01, 4.0649623967526478e-02, 1.1375526396986200e-03, 2.2743978290032441e-05, 3.5372519699820537e-07, 4.5013401478051409e-09, 4.8471099230041167e-11, 4.5236215845941172e-13,
-8.3155129940488270e-01, 2.5084238558307564e-01, 1.5135943334285990e-02, 4.6498071026480366e-04, 9.7572103430983264e-06, 1.5630927908732275e-07, 2.0293140870730405e-09, 2.2169470873901089e-11,
 1.0823936849879583e+00,-5.1793552316184588e-01, 8.5479999325103154e-02, 6.0274276960538189e-03, 1.9698772159630734e-04, 4.2798932300048642e-06, 7.0125367514469557e-08, 9.2495532481679587e-10,
-1.6003292081498042e+00, 9.3572008437456744e-01,-2.8903775197220921e-01, 3.0719422141331131e-02, 2.4842467506665036e-03, 8.5298230423324554e-05, 1.9057080222780615e-06, 3.1805669219869356e-08,
 2.5360492925243716e+00,-1.6618549884059748e+00, 6.8175743900683483e-01,-1.5317317484332565e-01, 1.1216217662382511e-02, 1.0449443151645107e-03, 3.7479832148836273e-05, 8.5775146127530156e-07,
-4.1979042809303468e+00, 2.9617097119186551e+00,-1.4384033392752202e+00, 4.5176420165333059e-01,-7.8712659639880292e-02, 4.0737799852723590e-03, 4.4526402563700929e-04, 1.6640594094197763e-05,
 7.1596139928490015e+00,-5.3199466984122505e+00, 2.8941457450346810e+00,-1.1132014382797852e+00, 2.8175996036651185e-01,-3.9609390353052297e-02, 1.4439024413086473e-03, 1.9135686163733944e-04,
-1.2479560691261252e+01, 9.6359151927183717e+00,-5.6900452873674521e+00, 2.5145553063612267e+00,-7.9852137202033435e-01, 1.6846968276036239e-01,-1.9622121217220369e-02, 4.8643505348642993e-04,
 8.3978798313602487e-01, 5.0278777219392702e-02, 1.8083628128130330e-03, 4.6476863475103180e-05, 9.2923322568682679e-07, 1.5202190761830370e-08, 2.1045690830412920e-10, 2.5251620877033690e-12,
-7.8950920591663221e-01, 2.3085612570115763e-01, 1.8331034193029110e-02, 7.2906676100230338e-04, 1.9735596202560045e-05, 4.0729134286398402e-07, 6.8072985305878819e-09, 9.5702320096276054e-11,
 1.0203653316177899e+00,-4.8180517154468228e-01, 7.4448856144035347e-02, 7.1383682338402050e-03, 3.0452034832227702e-04, 8.5671472756799931e-06, 1.8122485327130275e-07, 3.0816565735468356e-09,
-1.5021705031624721e+00, 8.7155951951330246e-01,-2.6288790382160415e-01, 2.4942296026930530e-02, 2.8728472818709697e-03, 1.2995004279077571e-04, 3.7745024736590657e-06, 8.1513344950616236e-08,
 2.3737300226757747e+00,-1.5475416231151957e+00, 6.2647708924308343e-01,-1.3596691350369175e-01, 8.2848120872041423e-03, 1.1779484503180245e-03, 5.6246332153980064e-05, 1.6806713266889449e-06,
-3.9212716457909704e+00, 2.7564363568358425e+00,-1.3270738865913767e+00, 4.0919127294341562e-01,-6.8060135616446013e-02, 2.6185550432223930e-03, 4.8832716024563305e-04, 2.4587330797104786e-05,
 6.6777080026268134e+00,-4.9482428298270840e+00, 2.6751631930917630e+00,-1.0168287304062604e+00, 2.5137767639308761e-01,-3.3288620054200239e-02, 7.3347603782468170e-04, 2.0370096152682481e-04,
-1.1625950832453897e+01, 8.9575876260971974e+00,-5.2645575632663526e+00, 2.3069716282416373e+00,-7.2179523438153681e-01, 1.4795127807168990e-01,-1.5987769391248810e-02, 1.4436127249575748e-04,
 8.0900813390195137e-01, 5.9136711998269177e-02, 2.5984248346443460e-03, 8.1602119988568529e-05, 1.9937440989837620e-06, 3.9861393778217388e-08, 6.7441007289210616e-10, 9.8894709565733370e-12,
-7.4987142190368217e-01, 2.1226494919214414e-01, 2.1105221236656466e-02, 1.0331478057727132e-03, 3.4296010003590145e-05, 8.6675378398143374e-07, 1.7728481386030997e-08, 3.0490347624195681e-10,
 9.6213637109582628e-01,-4.4815194233560052e-01, 6.4420647123664826e-02, 8.0315447299392495e-03, 4.2536232244204982e-04, 1.4732011752951923e-05, 3.8247590540029950e-07, 7.9708020723250485e-09,
-1.4102883134314268e+00, 8.1181116411665255e-01,-2.3886249721994149e-01, 1.9820368221598276e-02, 3.1523975640603673e-03, 1.7882312166910724e-04, 6.4212061047703017e-06, 1.7059208103855432e-07,
 2.2220994775480793e+00,-1.4411489334070891e+00, 5.7547918379956142e-01,-1.2038863985068156e-01, 5.7560011575772587e-03, 1.2575873380518436e-03, 7.6204399632299953e-05, 2.8278977444582265e-06,
-3.6632484109551684e+00, 2.5655015484561563e+00,-1.2241719410728060e+00, 3.7031171928339490e-01,-5.8563403681184699e-02, 1.4003428606463469e-03, 5.0574323979248991e-04, 3.2775061712515717e-05,
 6.2287499594113243e+00,-4.6026989794897499e+00, 2.4725595920253056e+00,-9.2842163898956009e-01, 2.2392435828527485e-01,-2.7744211857661669e-02, 1.5821695988948634e-04, 2.0390775615499368e-04,
-1.0831448938901074e+01, 8.3273220273103945e+00,-4.8706921672149415e+00, 2.1160468117348481e+00,-6.5196964886822473e-01, 1.2961431123818087e-01,-1.2853441333188338e-02,-1.2250678576580822e-04,
 7.7976817698039569e-01, 6.7277296171929499e-02, 3.4916826889013811e-03, 1.2955215459523910e-04, 3.7400651834043109e-06, 8.8359785089865186e-08, 1.7665834484750311e-09, 3.0612666113608771e-11,
-7.1249088080846623e-01, 1.9497321794746991e-01, 2.3496967072627564e-02, 1.3690204636101551e-03, 5.3887758961751879e-05, 1.6126334347021716e-06, 3.9031339567062386e-08, 7.9404259425038293e-10,
 9.0746409875593614e-01,-4.1680553350187360e-01, 5.5313732384526498e-02, 8.7319012869226578e-03, 5.5546168788426261e-04, 2.2903092278585263e-05, 7.0568857630776713e-07, 1.7428017873227283e-08,
-1.3242696322578098e+00, 7.5616938801020328e-01,-2.1679680501312235e-01, 1.5291390699522562e-02, 3.3381072686805768e-03, 2.2996441416497468e-04, 9.8743537157402464e-06, 3.1208523233204411e-07,
 2.0804390202680132e+00,-1.3421238021048882e+00, 5.2843939463691725e-01,-1.0629681726590039e-01, 3.5858243834211026e-03, 1.2927430827166003e-03, 9.6431114100888404e-05, 4.3001304820664107e-06,
-3.4225628223729014e+00, 2.3878964052855043e+00,-1.1290670058384129e+00, 3.3481988604367696e-01,-5.0110663677867602e-02, 3.8977588107529429e-04, 5.0253499433557829e-04, 4.0775245729729223e-05,
 5.8104592276584057e+00,-4.2814620166354134e+00, 2.2851174995788206e+00,-8.4734041064749255e-01, 1.9913604407565472e-01,-2.2893240283195705e-02,-3.0068253229251132e-04, 1.9474714709673377e-04,
-1.0091921244293818e+01, 7.7416934255740957e+00,-4.5061065526214819e+00, 1.9404685070816470e+00,-5.8845024896520459e-01, 1.1324561529997010e-01,-1.0160564519207264e-02,-3.2582032446300539e-04,
 7.5198060650995613e-01, 7.4750705123524219e-02, 4.4735552747135884e-03, 1.9145054496441701e-04, 6.3758922105259504e-06, 1.7377864863844500e-07, 4.0084271163009513e-09, 8.0140265403706879e-11,
-6.7722990138643191e-01, 1.7889186722960354e-01, 2.5541597101674748e-02, 1.7294307968902785e-03, 7.8809782578458508e-05, 2.7265405891186056e-06, 7.6240255475121822e-08, 1.7912131067983940e-09,
 8.5612176861603539e-01,-3.8760743629063099e-01, 4.7052808268300841e-02, 9.2620021224437323e-03, 6.9134087192228668e-04, 3.3137673449530779e-05, 1.1831275927140491e-06, 3.3806938667362403e-08,
-1.2437292049066664e+00, 7.0434989800817671e-01,-1.9653858151108700e-01, 1.1298539062212220e-02, 3.4435143344135558e-03, 2.8175351974753449e-04, 1.4129386966351814e-05, 5.1874964759009641e-07,
 1.9480791029148432e+00,-1.2499520206511234e+00, 4.8505766415168505e-01,-9.3561637233013181e-02, 1.7345883153182832e-03, 1.2911872069675465e-03, 1.1619002580428953e-04, 6.0830234475917119e-06,
-3.1980311235659666e+00, 2.2226834977238612e+00,-1.0411754547519423e+00, 3.0243468662106004e-01,-4.2600211970564392e-02,-4.3936159502053250e-04, 4.8301800542174280e-04, 4.8261228703185599e-05,
 5.4207146212898278e+00,-3.9828108420804780e+00, 2.1117096658141228e+00,-7.7299571692650071e-01, 1.7677153847157967e-01,-1.8660906752278696e-02,-6.5981346459104248e-04, 1.7855357219214476e-04,
-9.4035254633703058e+00, 7.1975221597198358e+00,-4.1686314328022149e+00, 1.7790264528305988e+00,-5.3069236085684590e-01, 9.8651707804128880e-02,-7.8567684975879709e-03,-4.7573153804546752e-04,
 7.2556324652119175e-01, 8.1603675475303103e-02, 5.5308158567526321e-03, 2.6814817011733050e-04, 1.0118236689945199e-05, 3.1249303054554872e-07, 8.1680546480394421e-09, 1.8505870874646271e-10,
-6.4395957104588863e-01, 1.6393795060292921e-01, 2.7271543235439007e-02, 2.1079833323137658e-03, 1.0923233504583809e-04, 4.2911607674193806e-06, 1.3615914490846191e-07, 3.6286733805526805e-09,
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-3.9456892574754562e-06,-2.4416146156498367e-06,-4.3681867962589667e-07, 9.1193072510117244e-07, 1.1438516959792810e-06, 6.1464166431485353e-07,-1.9116681760898808e-08,-3.4001824858238681e-07,
 1.5040746418256195e-06, 8.3517241007408694e-07, 7.3312684266654715e-09,-4.4550919009553901e-07,-3.9708156020384134e-07,-1.0513635619385240e-07, 1.1971070066954601e-07, 1.5990305720249418e-07,
-6.6890223175153253e-07,-3.2892668384777014e-07, 5.9432181545645940e-08, 2.2174741359048630e-07, 1.4066838894257268e-07,-1.0058516260868574e-08,-8.2133526573057944e-08,-6.0092826613545627e-08,
 3.1685678073510172e-02, 2.9677713683555120e-02, 2.6043527183100431e-02, 2.1425645501330651e-02, 1.6539121700887859e-02, 1.1992686356337569e-02, 8.1792009201709179e-03, 5.2544346477050004e-03,
-2.0079643899550519e-03,-1.7534695369847801e-03,-1.3170544088799939e-03,-8.1320716520883254e-04,-3.5400905788669779e-04,-1.3521444931679019e-05, 1.8519605459526786e-04, 2.6092107717821163e-04,
 2.5449485297027175e-04, 2.0611180107976690e-04, 1.2774229496078228e-04, 4.6464385467865060e-05,-1.4928262723716090e-05,-4.6375733419401155e-05,-5.0940756852390381e-05,-3.9192920056834878e-05,
-4.8383051890504847e-05,-3.6118653449154807e-05,-1.7418943248156477e-05,-2.4812346965327894e-07, 9.8148043653221394e-06, 1.1804292080044542e-05, 8.4326921666787600e-06, 3.4916534073899257e-06,
 1.2264398441350039e-05, 8.3783406535488738e-06, 2.8226077867025850e-06,-1.6086782850868457e-06,-3.3671414184766284e-06,-2.7433670958006101e-06,-1.1044362949867041e-06, 2.7363078286392646e-07,
-3.8860577878011655e-06,-2.4084693152971380e-06,-4.3647849633514288e-07, 8.9428652597273750e-07, 1.1280804607710246e-06, 6.1042847195484940e-07,-1.4393481533001610e-08,-3.3367001556636434e-07,
 1.4775884725040275e-06, 8.2213105513998769e-07, 9.6626857999302333e-09,-4.3673133253292754e-07,-3.9149285369711383e-07,-1.0551561607693006e-07, 1.1646122943796668e-07, 1.5751321419740637e-07,
-6.5545741736403985e-07,-3.2315977377202512e-07, 5.7150936736308337e-08, 2.1716228147740403e-07, 1.3877025252862932e-07,-8.9113740919580257e-09,-8.0345936905638683e-08,-5.9399626984098107e-08,
 3.1605562579013327e-02, 2.9607739407142591e-02, 2.5990947015079851e-02, 2.1393154383413170e-02, 1.6524949500295250e-02, 1.1992108523573901e-02, 8.1865680995801427e-03, 5.2648401737292868e-03,
-1.9978231718707357e-03,-1.7452538707515841e-03,-1.3119586221749108e-03,-8.1134880535308970e-04,-3.5459837234834077e-04,-1.5367060032345919e-05, 1.8316515869281312e-04, 2.5935615658193630e-04,
 2.5256930111915155e-04, 2.0467373201139838e-04, 1.2704779066242326e-04, 4.6453183079076107e-05,-1.4538352271002382e-05,-4.5905749929910897e-05,-5.0604332855930148e-05,-3.9053038558311106e-05,
-4.7895569107753166e-05,-3.5785437863399032e-05,-1.7306388010418242e-05,-3.1175980718178907e-07, 9.6810170117523729e-06, 1.1695044612584279e-05, 8.3885044338037569e-06, 3.5023333765164981e-06,
 1.2110131244354134e-05, 8.2826561538691498e-06, 2.8051569807495556e-06,-1.5732539048532446e-06,-3.3223299190641239e-06,-2.7190344615311516e-06,-1.1049197175636670e-06, 2.6040936105126460e-07,
-3.8274750904849842e-06,-2.3758410852514020e-06,-4.3604686211384259e-07, 8.7698979999498332e-07, 1.1125312605009309e-06, 6.0620101601273143e-07,-9.7988395000298255e-09,-3.2741682220803565e-07,
 1.4516340052335823e-06, 8.0931814701384549e-07, 1.1904308010271655e-08,-4.2813487775752673e-07,-3.8597967382942085e-07,-1.0584981005777915e-07, 1.1328255800927131e-07, 1.5515106083641173e-07,
-6.4231585821973676e-07,-3.1750393992913694e-07, 5.4942024140750536e-08, 2.1267691043083125e-07, 1.3689205590508720e-07,-7.8050100414459765e-09,-7.8593437585563066e-08,-5.8708386277910262e-08,
 3.1525851197981833e-02, 2.9538092610468261e-02, 2.5938570124122751e-02, 2.1360737590253021e-02, 1.6510754037630259e-02, 1.1991457241030300e-02, 8.1938543118210902e-03, 5.2751832149474428e-03,
-1.9877585875135717e-03,-1.7370954617258183e-03,-1.3068905257836334e-03,-8.0949094412597830e-04,-3.5517219694541566e-04,-1.7193962932315898e-05, 1.8114768439536459e-04, 2.5779683964921238e-04,
 2.5066312578775340e-04, 2.0324891536553885e-04, 1.2635777461771905e-04, 4.6439463392755056e-05,-1.4153757628527933e-05,-4.5440116918128715e-05,-5.0269676706857572e-05,-3.8912740371707297e-05,
-4.7414210422214555e-05,-3.5456023691141694e-05,-1.7194530435850474e-05,-3.7399291574856787e-07, 9.5490097374586688e-06, 1.1586767065057347e-05, 8.3443010060326941e-06, 3.5124894661959301e-06,
 1.1958186731072861e-05, 8.1882665931698626e-06, 2.7877252099446551e-06,-1.5385145639252532e-06,-3.2781359972039503e-06,-2.6948711811037562e-06,-1.1052218788238022e-06, 2.4743618463188650e-07,
-3.7699201379029987e-06,-2.3437208761824721e-06,-4.3552731103186154e-07, 8.6003336978499101e-07, 1.0972010904017509e-06, 6.0196106665261424e-07,-5.3299514992235271e-09,-3.2125756259483598e-07,
 1.4261992617205266e-06, 7.9672928952689830e-07, 1.4058982429867177e-08,-4.1971587999764545e-07,-3.8054122235696915e-07,-1.0614054456048576e-07, 1.1017329471090308e-07, 1.5281651378563946e-07,
-6.2946997219362833e-07,-3.1195688066681134e-07, 5.2803205382304908e-08, 2.0828903112074902e-07, 1.3503386204663466e-07,-6.7382139412803666e-09,-7.6875423238859881e-08,-5.8019355695892334e-08,
 3.1446540880502659e-02, 2.9468771013805350e-02, 2.5886395406199750e-02, 2.1328395099906049e-02, 1.6496535928253689e-02, 1.1990733253724710e-02, 8.2010600923435178e-03, 5.2854639958337842e-03,
-1.9777698666973084e-03,-1.7289937828379662e-03,-1.3018499407339811e-03,-8.0763368110405021e-04,-3.5573074289282913e-04,-1.9002326876569817e-05, 1.7914356097838667e-04, 2.5624314263109939e-04,
 2.4877608385934225e-04, 2.0183720011620948e-04, 1.2567221893636455e-04, 4.6423281993584597e-05,-1.3774408084843079e-05,-4.4978795722868864e-05,-4.9936788889288466e-05,-3.8772046254608827e-05,
-4.6938883743132768e-05,-3.5130359538852382e-05,-1.7083369693729987e-05,-4.3484992648155316e-07, 9.4187580137313058e-06, 1.1479452653782847e-05, 8.3000890339442375e-06, 3.5221315313995936e-06,
 1.1808524204280386e-05, 8.0951518286548015e-06, 2.7703159223001024e-06,-1.5044467915562664e-06,-3.2345509511770807e-06,-2.6708777199353983e-06,-1.1053477537523729e-06, 2.3470751830251741e-07,
-3.7133723756255840e-06,-2.3120998123612723e-06,-4.3492326577194122e-07, 8.4341021472654996e-07, 1.0820869771879632e-06, 5.9771033098997675e-07,-9.8406867030152053e-10,-3.1519113428088771e-07,
 1.4012725632643117e-06, 7.8436017663811695e-07, 1.6129469663362223e-08,-4.1147048452140296e-07,-3.7517669787062058e-07,-1.0638937881561243e-07, 1.0713207220673587e-07, 1.5050947977557964e-07,
-6.1691238662619474e-07,-3.0651634244110494e-07, 5.0732310279042822e-08, 2.0399642987963217e-07, 1.3319571651910126e-07,-5.7098081037228053e-09,-7.5191296396321809e-08,-5.7332773784924837e-08,
 3.1367628607276862e-02, 2.9399772358389621e-02, 2.5834421764418081e-02, 2.1296126886489328e-02, 1.6482295779134171e-02, 1.1989937299776191e-02, 8.2081859737679185e-03, 5.2956827414961972e-03,
-1.9678562488872411e-03,-1.7209483130186144e-03,-1.2968366891686345e-03,-8.0577711366205604e-04,-3.5627421859658545e-04,-2.0792323569067753e-05, 1.7715271770366199e-04, 2.5469508095619150e-04,
 2.4690793586862672e-04, 2.0043843727711105e-04, 1.2499109576395500e-04, 4.6404693391735412e-05,-1.3400233902699293e-05,-4.4521747954655462e-05,-4.9605669605282446e-05,-3.8630976573363464e-05,
-4.6469498591515668e-05,-3.4808394811598829e-05,-1.6972904818069319e-05,-4.9435743729715309e-07, 9.2902376575508833e-06, 1.1373094577819908e-05, 8.2558754713226760e-06, 3.5312692785416616e-06,
 1.1661103779916839e-05, 8.0032920690473949e-06, 2.7529324310514936e-06,-1.4710373947034785e-06,-3.1915661975607792e-06,-2.6470544763935028e-06,-1.1053022084987472e-06, 2.2221967082276520e-07,
-3.6578117108694440e-06,-2.2809691883741195e-06,-4.3423804025462227e-07, 8.2711346654795919e-07, 1.0671859796961762e-06, 5.9345045468474097e-07, 3.2415036832349010e-09,-3.0921643880795001e-07,
 1.3768425224953245e-06, 7.7220659124789020e-07, 1.8118444833749962e-08,-4.0339492397355387e-07,-3.6988529751667154e-07,-1.0659782554107820e-07, 1.0415754681313402e-07, 1.4822985635144091e-07,
-6.0463593124743431e-07,-3.0118012052694874e-07, 4.8727237281273830e-08, 1.9979694460706299e-07, 1.3137765045004862e-07,-4.7186477956994096e-09,-7.3540466719778852e-08,-5.6648867394321842e-08,
 3.1289111389236159e-02, 2.9331094406180890e-02, 2.5782648108976899e-02, 2.1263932920268219e-02, 1.6468034188959531e-02, 1.1989070110466229e-02, 8.2152324858851596e-03, 5.3058396776434974e-03,
-1.9580169830552691e-03,-1.7129585371175732e-03,-1.2918505943436097e-03,-8.0392133701552917e-04,-3.5680282969241875e-04,-2.2564123183627452e-05, 1.7517508383036763e-04, 2.5315266924616844e-04,
 2.4505844593769588e-04, 1.9905247987007718e-04, 1.2431437728726278e-04, 4.6383751044028611e-05,-1.3031166305387204e-05,-4.4068935496362354e-05,-4.9276318782566513e-05,-3.8489551309035916e-05,
-4.6005966067618701e-05,-3.4490079699336706e-05,-1.6863134712814733e-05,-5.5254152399439229e-07, 9.1634248269763862e-06, 1.1267686021604621e-05, 8.2116670793799343e-06, 3.5399122673234788e-06,
 1.1515886368281995e-05, 7.9126678682539849e-06, 2.7355779186834153e-06,-1.4382734517968435e-06,-3.1491732700824367e-06,-2.6234017842263000e-06,-1.1050900023882356e-06, 2.0996899470191341e-07,
-3.6032185000280100e-06,-2.2503204663710434e-06,-4.3347484245992721e-07, 8.1113640547120060e-07, 1.0524951891581172e-06, 5.8918302379524593e-07, 7.3494068155889523e-09,-3.0333238181259276e-07,
 1.3528980336569665e-06, 7.6026440472175529e-07, 2.0028499194230117e-08,-3.9548551557795952e-07,-3.6466621795133588e-07,-1.0676735227850774e-07, 1.0124839857543617e-07, 1.4597753190004634e-07,
-5.9263362893521126e-07,-2.9594606070661304e-07, 4.6785953347696299e-08, 1.9568846211758051e-07, 1.2957968198418289e-07,-3.7636202945888255e-09,-7.1922351680876334e-08,-5.5967851324670247e-08,
};

// rs, ws = dft.radi.gauss_chebyshev(nrs)
static double rs_gauss_chebyshev2047[] = {
2.109423746787797e-15, 6.672440377997191e-14, 5.055955654142963e-13, 2.130295939650750e-12, 6.500799898390142e-12,
1.617572742418361e-11, 3.496158917926095e-11, 6.816192055225656e-11, 1.228277479725648e-10, 2.080052796671339e-10,
3.349868160640312e-10, 5.175581074823299e-10, 7.722494865802787e-10, 1.118582004799862e-09, 1.579318786859574e-09,
2.180712166399701e-09, 2.952752042872930e-09, 3.929414571501866e-09, 5.148910409147334e-09, 6.653931405864455e-09,
8.491899960993976e-09, 1.071521438245071e-08, 1.338149835383717e-08, 1.655384551657590e-08, 2.030106827088929e-08,
2.469794224690958e-08, 2.982545377339108e-08, 3.577104545904319e-08, 4.262886077466277e-08, 5.049999096673474e-08,
5.949271808525225e-08, 6.972276078709427e-08, 8.131351858509106e-08, 9.439631398766579e-08, 1.091106383022122e-07,
1.256043927755357e-07, 1.440341329539407e-07, 1.645653087134491e-07, 1.873725090639766e-07, 2.126397006252390e-07,
2.405604703215047e-07, 2.713382656338581e-07, 3.051866345193943e-07, 3.423294646642816e-07, 3.830012232919344e-07,
4.274471947507408e-07, 4.759237197671240e-07, 5.286984324781585e-07, 5.860504967980518e-07, 6.482708438948492e-07,
7.156624082238494e-07, 7.885403613405728e-07, 8.672323489333778e-07, 9.520787229710947e-07, 1.043432776848263e-06,
1.141660977199699e-06, 1.247143197269374e-06, 1.360272948169872e-06, 1.481457609364689e-06, 1.611118660815869e-06,
1.749691911134832e-06, 1.897627727176499e-06, 2.055391262745232e-06, 2.223462686079536e-06, 2.402337407003685e-06,
2.592526303746290e-06, 2.794555946650235e-06, 3.008968825213287e-06, 3.236323571464972e-06, 3.477195182233217e-06,
3.732175243409408e-06, 4.001872150327657e-06, 4.286911329698384e-06, 4.587935458988390e-06, 4.905604686689102e-06,
5.240596849920287e-06, 5.593607693366032e-06, 5.965351084213921e-06, 6.356559230646930e-06, 6.767982894895219e-06,
7.200391608730428e-06, 7.654573886295424e-06, 8.131337437489172e-06, 8.631509378020930e-06, 9.155936441795909e-06,
9.705485189082097e-06, 1.028104221745263e-05, 1.088351436739909e-05, 1.151382893083142e-05, 1.217293385680218e-05,
1.286179795578768e-05, 1.358141110463507e-05, 1.433278444951114e-05, 1.511695060718576e-05, 1.593496386720350e-05,
1.678790039139066e-05, 1.767685841225219e-05, 1.860295843236770e-05, 1.956734342012378e-05, 2.057117900589045e-05,
2.161565367753138e-05, 2.270197897313864e-05, 2.383138967454457e-05, 2.500514399861320e-05, 2.622452378708839e-05,
2.749083469755220e-05, 2.880540638994233e-05, 3.016959271540109e-05, 3.158477190057241e-05, 3.305234673423030e-05,
3.457374474935548e-05, 3.615041840676625e-05, 3.778384527719503e-05, 3.947552821959022e-05, 4.122699556186049e-05,
4.303980127806639e-05, 4.491552516561192e-05, 4.685577301977162e-05, 4.886217680932781e-05, 5.093639484854418e-05,
5.308011196969442e-05, 5.529503969314842e-05, 5.758291639734736e-05, 5.994550748600336e-05, 6.238460555652026e-05,
6.490203056430666e-05, 6.749962998708892e-05, 7.017927898900211e-05, 7.294288058168341e-05, 7.579236578481030e-05,
7.872969378541761e-05, 8.175685209610428e-05, 8.487585671101971e-05, 8.808875226173907e-05, 9.139761217080711e-05,
9.480453880483797e-05, 9.831166362528343e-05, 1.019211473387571e-04, 1.056351800459154e-04, 1.094559813878959e-04,
1.133858006934219e-04, 1.174269171216880e-04, 1.215816398080216e-04, 1.258523080027718e-04, 1.302412912136397e-04,
1.347509893450116e-04, 1.393838328340724e-04, 1.441422827881400e-04, 1.490288311192245e-04, 1.540460006780320e-04,
1.591963453845269e-04, 1.644824503607145e-04, 1.699069320578728e-04, 1.754724383852269e-04, 1.811816488362927e-04,
1.870372746145543e-04, 1.930420587546999e-04, 1.991987762471892e-04, 2.055102341569359e-04, 2.119792717443225e-04,
2.186087605804410e-04, 2.254016046659979e-04, 2.323607405440020e-04, 2.394891374138952e-04, 2.467897972433519e-04,
2.542657548791905e-04, 2.619200781547315e-04, 2.697558679988221e-04, 2.777762585403076e-04, 2.859844172141690e-04,
2.943835448622201e-04, 3.029768758371354e-04, 3.117676780998169e-04, 3.207592533200909e-04, 3.299549369719657e-04,
3.393580984301092e-04, 3.489721410631086e-04, 3.588005023260621e-04, 3.688466538522839e-04, 3.791141015414556e-04,
3.896063856482224e-04, 4.003270808684567e-04, 4.112797964238579e-04, 4.224681761463289e-04, 4.338958985578012e-04,
4.455666769520583e-04, 4.574842594735617e-04, 4.696524291936122e-04, 4.820750041871769e-04, 4.947558376058314e-04,
5.076988177517006e-04, 5.209078681476242e-04, 5.343869476069907e-04, 5.481400503023481e-04, 5.621712058306860e-04,
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3.670643710673850e+00, 3.680566170400736e+00, 3.690511534739940e+00, 3.700479868780675e+00, 3.710471237933548e+00,
3.720485707932715e+00, 3.730523344838058e+00, 3.740584215037369e+00, 3.750668385248565e+00, 3.760775922521908e+00,
3.770906894242247e+00, 3.781061368131299e+00, 3.791239412249907e+00, 3.801441095000366e+00, 3.811666485128723e+00,
3.821915651727133e+00, 3.832188664236210e+00, 3.842485592447412e+00, 3.852806506505434e+00, 3.863151476910643e+00,
3.873520574521499e+00, 3.883913870557034e+00, 3.894331436599327e+00, 3.904773344596001e+00, 3.915239666862761e+00,
3.925730476085937e+00, 3.936245845325039e+00, 3.946785848015371e+00, 3.957350557970621e+00, 3.967940049385509e+00,
3.978554396838440e+00, 3.989193675294189e+00, 3.999857960106600e+00, 4.010547327021312e+00, 4.021261852178517e+00,
4.032001612115733e+00, 4.042766683770594e+00, 4.053557144483683e+00, 4.064373072001384e+00, 4.075214544478740e+00,
4.086081640482366e+00, 4.096974438993361e+00, 4.107893019410271e+00, 4.118837461552051e+00, 4.129807845661077e+00,
4.140804252406161e+00, 4.151826762885632e+00, 4.162875458630386e+00, 4.173950421607024e+00, 4.185051734220966e+00,
4.196179479319638e+00, 4.207333740195653e+00, 4.218514600590035e+00, 4.229722144695478e+00, 4.240956457159616e+00,
4.252217623088338e+00, 4.263505728049137e+00, 4.274820858074468e+00, 4.286163099665153e+00, 4.297532539793814e+00,
4.308929265908342e+00, 4.320353365935387e+00, 4.331804928283875e+00, 4.343284041848587e+00, 4.354790796013735e+00,
4.366325280656599e+00, 4.377887586151166e+00, 4.389477803371849e+00, 4.401096023697188e+00, 4.412742339013631e+00,
4.424416841719308e+00, 4.436119624727885e+00, 4.447850781472409e+00, 4.459610405909228e+00, 4.471398592521917e+00,
4.483215436325263e+00, 4.495061032869268e+00, 4.506935478243211e+00, 4.518838869079734e+00, 4.530771302558964e+00,
4.542732876412686e+00, 4.554723688928536e+00, 4.566743838954274e+00, 4.578793425902044e+00, 4.590872549752711e+00,
4.602981311060233e+00, 4.615119810956068e+00, 4.627288151153625e+00, 4.639486433952758e+00, 4.651714762244315e+00,
4.663973239514709e+00, 4.676261969850549e+00, 4.688581057943313e+00, 4.700930609094060e+00, 4.713310729218199e+00,
4.725721524850293e+00, 4.738163103148915e+00, 4.750635571901562e+00, 4.763139039529587e+00, 4.775673615093210e+00,
4.788239408296574e+00, 4.800836529492834e+00, 4.813465089689299e+00, 4.826125200552656e+00, 4.838816974414199e+00,
4.851540524275148e+00, 4.864295963811994e+00, 4.877083407381912e+00, 4.889902970028242e+00, 4.902754767485968e+00,
4.915638916187349e+00, 4.928555533267491e+00, 4.941504736570079e+00, 4.954486644653096e+00, 4.967501376794639e+00,
4.980549052998763e+00, 4.993629794001427e+00, 5.006743721276453e+00, 5.019890957041582e+00, 5.033071624264576e+00,
5.046285846669371e+00, 5.059533748742330e+00, 5.072815455738514e+00, 5.086131093688055e+00, 5.099480789402581e+00,
5.112864670481698e+00, 5.126282865319549e+00, 5.139735503111452e+00, 5.153222713860579e+00, 5.166744628384741e+00,
5.180301378323200e+00, 5.193893096143594e+00, 5.207519915148915e+00, 5.221181969484560e+00, 5.234879394145455e+00,
5.248612324983266e+00, 5.262380898713673e+00, 5.276185252923733e+00, 5.290025526079311e+00, 5.303901857532590e+00,
5.317814387529677e+00, 5.331763257218268e+00, 5.345748608655417e+00, 5.359770584815368e+00, 5.373829329597493e+00,
5.387924987834285e+00, 5.402057705299476e+00, 5.416227628716221e+00, 5.430434905765360e+00, 5.444679685093800e+00,
5.458962116322963e+00, 5.473282350057335e+00, 5.487640537893116e+00, 5.502036832426946e+00, 5.516471387264770e+00,
5.530944357030720e+00, 5.545455897376190e+00, 5.560006164988958e+00, 5.574595317602406e+00, 5.589223514004851e+00,
5.603890914049015e+00, 5.618597678661549e+00, 5.633343969852670e+00, 5.648129950725965e+00, 5.662955785488214e+00,
5.677821639459424e+00, 5.692727679082864e+00, 5.707674071935328e+00, 5.722660986737433e+00, 5.737688593364079e+00,
5.752757062854974e+00, 5.767866567425386e+00, 5.783017280476891e+00, 5.798209376608307e+00, 5.813443031626794e+00,
5.828718422558992e+00, 5.844035727662364e+00, 5.859395126436613e+00, 5.874796799635312e+00, 5.890240929277553e+00,
5.905727698659847e+00, 5.921257292368080e+00, 5.936829896289670e+00, 5.952445697625814e+00, 5.968104884903939e+00,
5.983807647990229e+00, 5.999554178102374e+00, 6.015344667822389e+00, 6.031179311109692e+00, 6.047058303314223e+00,
6.062981841189806e+00, 6.078950122907638e+00, 6.094963348069916e+00, 6.111021717723705e+00, 6.127125434374859e+00,
6.143274702002223e+00, 6.159469726071964e+00, 6.175710713552025e+00, 6.191997872926832e+00, 6.208331414212177e+00,
6.224711548970203e+00, 6.241138490324660e+00, 6.257612452976330e+00, 6.274133653218585e+00, 6.290702308953236e+00,
6.307318639706492e+00, 6.323982866645138e+00, 6.340695212592962e+00, 6.357455902047350e+00, 6.374265161196075e+00,
6.391123217934304e+00, 6.408030301881873e+00, 6.424986644400687e+00, 6.441992478612433e+00, 6.459048039416443e+00,
6.476153563507849e+00, 6.493309289395911e+00, 6.510515457422603e+00, 6.527772309781480e+00, 6.545080090536701e+00,
6.562439045642397e+00, 6.579849422962180e+00, 6.597311472288993e+00, 6.614825445365213e+00, 6.632391595902918e+00,
6.650010179604594e+00, 6.667681454183917e+00, 6.685405679386927e+00, 6.703183117013510e+00, 6.721014030939050e+00,
6.738898687136427e+00, 6.756837353698344e+00, 6.774830300859905e+00, 6.792877801021459e+00, 6.810980128771898e+00,
6.829137560912018e+00, 6.847350376478465e+00, 6.865618856767845e+00, 6.883943285361148e+00, 6.902323948148620e+00,
6.920761133354844e+00, 6.939255131564280e+00, 6.957806235747054e+00, 6.976414741285144e+00, 6.995080945998938e+00,
7.013805150174196e+00, 7.032587656589204e+00, 7.051428770542594e+00, 7.070328799881272e+00, 7.089288055028900e+00,
7.108306849014756e+00, 7.127385497502940e+00, 7.146524318822049e+00, 7.165723633995188e+00, 7.184983766770552e+00,
7.204305043652314e+00, 7.223687793931977e+00, 7.243132349720228e+00, 7.262639045979126e+00, 7.282208220554966e+00,
7.301840214211342e+00, 7.321535370662962e+00, 7.341294036609699e+00, 7.361116561771329e+00, 7.381003298922700e+00,
7.400954603929380e+00, 7.420970835783875e+00, 7.441052356642428e+00, 7.461199531862184e+00, 7.481412730039120e+00,
7.501692323046366e+00, 7.522038686073230e+00, 7.542452197664677e+00, 7.562933239761500e+00, 7.583482197741078e+00,
7.604099460458579e+00, 7.624785420289157e+00, 7.645540473170391e+00, 7.666365018645576e+00, 7.687259459907600e+00,
7.708224203843671e+00, 7.729259661080338e+00, 7.750366246029681e+00, 7.771544376935775e+00, 7.792794475922316e+00,
7.814116969040460e+00, 7.835512286317934e+00, 7.856980861808525e+00, 7.878523133642516e+00, 7.900139544077947e+00,
7.921830539552562e+00, 7.943596570736623e+00, 7.965438092586653e+00, 7.987355564399849e+00, 8.009349449869553e+00,
8.031420217141484e+00, 8.053568338870971e+00, 8.075794292280927e+00, 8.098098559221048e+00, 8.120481626227640e+00,
8.142943984584740e+00, 8.165486130385943e+00, 8.188108564597510e+00, 8.210811793122247e+00, 8.233596326864642e+00,
8.256462681797082e+00, 8.279411379026904e+00, 8.302442944864886e+00, 8.325557910894734e+00, 8.348756814043647e+00,
8.372040196654362e+00, 8.395408606558021e+00, 8.418862597148458e+00, 8.442402727457974e+00, 8.466029562234013e+00,
8.489743672017255e+00, 8.513545633221351e+00, 8.537436028213545e+00, 8.561415445397042e+00, 8.585484479294841e+00,
8.609643730634522e+00, 8.633893806435257e+00, 8.658235320095770e+00, 8.682668891483992e+00, 8.707195147028431e+00,
8.731814719811089e+00, 8.756528249661875e+00, 8.781336383255059e+00, 8.806239774206903e+00, 8.831239083175706e+00,
8.856334977963076e+00, 8.881528133617271e+00, 8.906819232538691e+00, 8.932208964586735e+00, 8.957698027188872e+00,
8.983287125451987e+00, 9.008976972275276e+00, 9.034768288465537e+00, 9.060661802854733e+00, 9.086658252419237e+00,
9.112758382401763e+00, 9.138962946435646e+00, 9.165272706670864e+00, 9.191688433903026e+00, 9.218210907704670e+00,
9.244840916558999e+00, 9.271579257996139e+00, 9.298426738731981e+00, 9.325384174809844e+00, 9.352452391744864e+00,
9.379632224671093e+00, 9.406924518491431e+00, 9.434330128030593e+00, 9.461849918191293e+00, 9.489484764112710e+00,
9.517235551333323e+00, 9.545103175955768e+00, 9.573088544815768e+00, 9.601192575654281e+00, 9.629416197292837e+00,
9.657760349812646e+00, 9.686225984737575e+00, 9.714814065220347e+00, 9.743525566232606e+00, 9.772361474759579e+00,
9.801322789997847e+00, 9.830410523557575e+00, 9.859625699669385e+00, 9.888969355394741e+00, 9.918442540841205e+00,
9.948046319382158e+00, 9.977781767881448e+00, 1.000764997692218e+01, 1.003765205104090e+01, 1.006778910896657e+01,
1.009806228386475e+01, 1.012847272358711e+01, 1.015902159092605e+01, 1.018971006387557e+01, 1.022053933589693e+01,
1.025151061619078e+01, 1.028262512997506e+01, 1.031388411876938e+01, 1.034528884068487e+01, 1.037684057072195e+01,
1.040854060107327e+01, 1.044039024143443e+01, 1.047239081932158e+01, 1.050454368039601e+01, 1.053685018879641e+01,
1.056931172747886e+01, 1.060192969856399e+01, 1.063470552369310e+01, 1.066764064439239e+01, 1.070073652244486e+01,
1.073399464027219e+01, 1.076741650132477e+01, 1.080100363048166e+01, 1.083475757445879e+01, 1.086867990222930e+01,
1.090277220545159e+01, 1.093703609890872e+01, 1.097147322095934e+01, 1.100608523399792e+01, 1.104087382492768e+01,
1.107584070564463e+01, 1.111098761353359e+01, 1.114631631197722e+01, 1.118182859087630e+01, 1.121752626718531e+01,
1.125341118545871e+01, 1.128948521841485e+01, 1.132575026751078e+01, 1.136220826353296e+01, 1.139886116720549e+01,
1.143571096980984e+01, 1.147275969382474e+01, 1.151000939358083e+01, 1.154746215593246e+01, 1.158512010094877e+01,
1.162298538262129e+01, 1.166106018959245e+01, 1.169934674590217e+01, 1.173784731175548e+01, 1.177656418431080e+01,
1.181549969848945e+01, 1.185465622780946e+01, 1.189403618523908e+01, 1.193364202407642e+01, 1.197347623885395e+01,
1.201354136626696e+01, 1.205383998612883e+01, 1.209437472235599e+01, 1.213514824397457e+01, 1.217616326616551e+01,
1.221742255133045e+01, 1.225892891019650e+01, 1.230068520294748e+01, 1.234269434039220e+01, 1.238495928516590e+01,
1.242748305296925e+01, 1.247026871384086e+01, 1.251331939347148e+01, 1.255663827455916e+01, 1.260022859820111e+01,
1.264409366533197e+01, 1.268823683820882e+01, 1.273266154193326e+01, 1.277737126603530e+01, 1.282236956609193e+01,
1.286766006541033e+01, 1.291324645675497e+01, 1.295913250413619e+01, 1.300532204465481e+01, 1.305181899040709e+01,
1.309862733044861e+01, 1.314575113283309e+01, 1.319319454670351e+01, 1.324096180446875e+01, 1.328905722404351e+01,
1.333748521116541e+01, 1.338625026179792e+01, 1.343535696460546e+01, 1.348481000352272e+01, 1.353461416040978e+01,
1.358477431779985e+01, 1.363529546174767e+01, 1.368618268477408e+01, 1.373744118891886e+01, 1.378907628890699e+01,
1.384109341542250e+01, 1.389349811851252e+01, 1.394629607110189e+01, 1.399949307265937e+01, 1.405309505297676e+01,
1.410710807611295e+01, 1.416153834447302e+01, 1.421639220305289e+01, 1.427167614383876e+01, 1.432739681038808e+01,
1.438356100257590e+01, 1.444017568155371e+01, 1.449724797487072e+01, 1.455478518183584e+01, 1.461279477907283e+01,
1.467128442631195e+01, 1.473026197241717e+01, 1.478973546166729e+01, 1.484971314029506e+01, 1.491020346330665e+01,
1.497121510158665e+01, 1.503275694931317e+01, 1.509483813168130e+01, 1.515746801299181e+01, 1.522065620506818e+01,
1.528441257604393e+01, 1.534874725958989e+01, 1.541367066449597e+01, 1.547919348473382e+01, 1.554532670997873e+01,
1.561208163660320e+01, 1.567946987918948e+01, 1.574750338260948e+01, 1.581619443467032e+01, 1.588555567934477e+01,
1.595560013069564e+01, 1.602634118744881e+01, 1.609779264831570e+01, 1.616996872804772e+01, 1.624288407436192e+01,
1.631655378568588e+01, 1.639099342984711e+01, 1.646621906374661e+01, 1.654224725402799e+01, 1.661909509891309e+01,
1.669678025119957e+01, 1.677532094246386e+01, 1.685473600869450e+01, 1.693504491725503e+01, 1.701626779547523e+01,
1.709842546078591e+01, 1.718153945262697e+01, 1.726563206626724e+01, 1.735072638856963e+01, 1.743684633586383e+01,
1.752401669422101e+01, 1.761226316217449e+01, 1.770161239597318e+01, 1.779209205780555e+01, 1.788373086712009e+01,
1.797655865508622e+01, 1.807060642285958e+01, 1.816590640344814e+01, 1.826249212791489e+01, 1.836039849606064e+01,
1.845966185186158e+01, 1.856032006432152e+01, 1.866241261403591e+01, 1.876598068591657e+01, 1.887106726862854e+01,
1.897771726169035e+01, 1.908597759027450e+01, 1.919589732891547e+01, 1.930752783498753e+01, 1.942092289218180e+01,
1.953613886627202e+01, 1.965323487269077e+01, 1.977227295834986e+01, 1.989331829915240e+01, 2.001643941341685e+01,
2.014170839483397e+01, 2.026920116529554e+01, 2.039899775129894e+01, 2.053118258471455e+01, 2.066584483244346e+01,
2.080307875718952e+01, 2.094298411360127e+01, 2.108566658297434e+01, 2.123123825214442e+01, 2.137981814251149e+01,
2.153153279315106e+01, 2.168651690873131e+01, 2.184491407636305e+01, 2.200687756536829e+01, 2.217257121614261e+01,
2.234217043696705e+01, 2.251586331910539e+01, 2.269385188821027e+01, 2.287635351670385e+01, 2.306360251909887e+01,
2.325585195976869e+01, 2.345337570600628e+01, 2.365647077640220e+01, 2.386546002269716e+01, 2.408069521580909e+01,
2.430256060899734e+01, 2.453147706166365e+01, 2.476790684179502e+01, 2.501235924363743e+01, 2.526539718737621e+01,
2.552764503779376e+01, 2.579979785401857e+01, 2.608263251337429e+01, 2.637702106424066e+01, 2.668394692894125e+01,
2.700452472831961e+01, 2.734002470651923e+01, 2.769190305493776e+01, 2.806184022922345e+01, 2.845178925765459e+01,
2.886403827217789e+01, 2.930129173049771e+01, 2.976677786933950e+01, 3.026439315807712e+01, 3.079890061034902e+01,
3.137620667940138e+01, 3.200375858512068e+01, 3.269112884843310e+01, 3.345090536982737e+01, 3.430009548529412e+01,
3.526242585320269e+01, 3.637228801250404e+01, 3.768170434501420e+01, 3.927176058857470e+01, 4.125684860588751e+01,
4.357373524529790e+01, 4.484012866322161e+01, 0.,
};
static double ws_gauss_chebyshev2047[] = {
1.040141248065865e-14, 1.664218164750223e-13, 8.425038375613978e-13, 2.662698938341439e-12, 6.500638049218187e-12,
1.347949045313434e-11, 2.497191092593100e-11, 4.259997513447332e-11, 6.823510329518064e-11, 1.039979720960954e-10,
1.522584148861276e-10, 2.156353101356187e-10, 2.969964581728792e-10, 3.994584462018767e-10, 5.263865016067688e-10,
6.813943335495081e-10, 8.683439628672716e-10, 1.091345540277019e-09, 1.354757152895011e-09, 1.663184619079679e-09,
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4.400355444273073e-02, 4.428336213519293e-02, 4.456662687673602e-02, 4.485341389631300e-02, 4.514379007413232e-02,
4.543782399442618e-02, 4.573558599994006e-02, 4.603714824871848e-02, 4.634258477304224e-02, 4.665197154069028e-02,
4.696538651850100e-02, 4.728290973878899e-02, 4.760462336791700e-02, 4.793061177813619e-02, 4.826096162201311e-02,
4.859576190993245e-02, 4.893510409101161e-02, 4.927908213701429e-02, 4.962779263004886e-02, 4.998133485382108e-02,
5.033981088872321e-02, 5.070332571109194e-02, 5.107198729654266e-02, 5.144590672785330e-02, 5.182519830765808e-02,
5.220997967580562e-02, 5.260037193226379e-02, 5.299649976494240e-02, 5.339849158400164e-02, 5.380647966132648e-02,
5.422060027715941e-02, 5.464099387290190e-02, 5.506780521131065e-02, 5.550118354380304e-02, 5.594128278599803e-02,
5.638826170099771e-02, 5.684228409236822e-02, 5.730351900500930e-02, 5.777214093711701e-02, 5.824833006154496e-02,
5.873227245843550e-02, 5.922416035927789e-02, 5.972419240331855e-02, 6.023257390671135e-02, 6.074951714552922e-02,
6.127524165309805e-02, 6.180997453292713e-02, 6.235395078742177e-02, 6.290741366510025e-02, 6.347061502504725e-02,
6.404381572070365e-02, 6.462728600642950e-02, 6.522130596345847e-02, 6.582616595121359e-02, 6.644216708351365e-02,
6.706962173067801e-02, 6.770885405015876e-02, 6.836020054851545e-02, 6.902401067524323e-02, 6.970064744995147e-02,
7.039048812873928e-02, 7.109392490838246e-02, 7.181136567395162e-02, 7.254323478986174e-02, 7.328997394213939e-02,
7.405204303036350e-02, 7.482992111660267e-02, 7.562410743459462e-02, 7.643512246096959e-02, 7.726350905876482e-02,
7.810983369459748e-02, 7.897468773330364e-02, 7.985868882391924e-02, 8.076248237350624e-02, 8.168674312725339e-02,
8.263217685253897e-02, 8.359952214218136e-02, 8.458955234763808e-02, 8.560307764709922e-02, 8.664094726097721e-02,
8.770405183579809e-02, 8.879332600333216e-02, 8.990975112426186e-02, 9.105435824770836e-02, 9.222823130012905e-02,
9.343251051187931e-02, 9.466839612987468e-02, 9.593715241066648e-02, 9.724011194914539e-02, 9.857868036165565e-02,
9.995434135119849e-02, 1.013686622100244e-01, 1.028232997924972e-01, 1.043200070026139e-01, 1.058606398495997e-01,
1.074471651569439e-01, 1.090816689466604e-01, 1.107663656076400e-01, 1.125036079374191e-01, 1.142958981007997e-01,
1.161458997119761e-01, 1.180564510381272e-01, 1.200305795535312e-01, 1.220715180084827e-01, 1.241827220798729e-01,
1.263678899607639e-01, 1.286309839853891e-01, 1.309762546828021e-01, 1.334082674145709e-01, 1.359319320967154e-01,
1.385525363255699e-01, 1.412757824333732e-01, 1.441078289265746e-01, 1.470553370223640e-01, 1.501255230791583e-01,
1.533262175560458e-01, 1.566659318789719e-01, 1.601539339820259e-01, 1.638003344031104e-01, 1.676161840762574e-01,
1.716135864704616e-01, 1.758058259373194e-01, 1.802075151350041e-01, 1.848347654315720e-01, 1.897053841569125e-01,
1.948391038040393e-01, 2.002578490946182e-01, 2.059860506445547e-01, 2.120510130261529e-01, 2.184833502462140e-01,
2.253175032175642e-01, 2.325923566075963e-01, 2.403519793681240e-01, 2.486465186036903e-01, 2.575332842542348e-01,
2.670780780746819e-01, 2.773568211297051e-01, 2.884575825808999e-01, 3.004831077874051e-01, 3.135540067457522e-01,
3.278128156815317e-01, 3.434292159604898e-01, 3.606068021496177e-01, 3.795920373061096e-01, 4.006860971742861e-01,
4.242609931230809e-01, 4.507816611158829e-01, 4.808368643715972e-01, 5.151832674168473e-01, 5.548098809961413e-01,
6.010342738416996e-01, 6.556508252475161e-01, 7.211660049076284e-01, 8.011867377877530e-01, 9.010874494589812e-01,
1.029199766035716e+00, 1.199011029546746e+00, 1.433109190324988e+00, 1.767258115946684e+00, 2.213710600108230e+00,
2.178771019099798e+00, 3.275790025202306e-01, 0.,
};

static double _factorial[] = {
        1.0, 1.0, 2.0, 6.0, 24.,
        1.2e+2, 7.2e+2, 5.04e+3, 4.032e+4, 3.6288e+5,
        3.6288e+6, 3.99168e+7, 4.790016e+8, 6.2270208e+9, 8.71782912e+10,
        1.307674368e+12, 2.0922789888e+13, 3.55687428096e+14,
        6.402373705728e+15, 1.21645100408832e+17,
        2.43290200817664e+18, 5.109094217170944e+19,
        1.1240007277776077e+21, 2.5852016738884978e+22,
};

const static double _factorial2[] = {
        1., 1., 2., 3., 8.,
        15., 48., 105., 384., 945.,
        3840., 10395., 46080., 135135., 645120.,
        2027025., 10321920., 34459425., 185794560., 654729075.,
        3715891200., 13749310575., 81749606400., 316234143225., 1961990553600.,
        7905853580625., 51011754393600., 213458046676875.,
        1428329123020800., 6190283353629376.,
        42849873690624000., 1.9189878396251069e+17,
        1.371195958099968e+18, 6.3326598707628524e+18,
        4.6620662575398912e+19, 2.2164309547669976e+20,
        1.6783438527143608e+21, 8.2007945326378929e+21,
        6.3777066403145712e+22, 3.1983098677287775e+23,
};

static double factorial2(int n)
{
        if (n < 0) {
                return 1;
        } else {
                return _factorial2[n];
        }
}

const static double _binom[] = {
        1,
        1, 1,
        1, 2, 1,
        1, 3, 3, 1,
        1, 4, 6, 4, 1,
        1, 5, 10, 10, 5, 1,
        1, 6, 15, 20, 15, 6, 1,
        1, 7, 21, 35, 35, 21, 7, 1,
        1, 8, 28, 56, 70, 56, 28, 8, 1,
        1, 9, 36, 84, 126, 126, 84, 36, 9, 1,};
static double binom(int n, int m)
{
        if (n < 10) {
                return _binom[n*(n+1)/2+m];
        } else {
                return _factorial[n] / (_factorial[m]*_factorial[n-m]);
        }
}

static int _cart_pow_y[] = {
        0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1,
        0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5,
        4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,10, 9, 8, 7, 6,
        5, 4, 3, 2, 1, 0,11,10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,12,11,
       10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,13,12,11,10, 9, 8, 7, 6, 5,
        4, 3, 2, 1, 0,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,
};
static int _cart_pow_z[] = {
        0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4,
        5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3,
        4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4,
        5, 6, 7, 8, 9,10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11, 0, 1,
        2, 3, 4, 5, 6, 7, 8, 9,10,11,12, 0, 1, 2, 3, 4, 5, 6, 7, 8,
        9,10,11,12,13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,
};
#define LOOP_CART(l, i, ix, iy, iz) \
        for (i = 0; iy = _cart_pow_y[i], \
                    iz = _cart_pow_z[i], \
                    ix = l - iy - iz, \
                    i < (l+1)*(l+2)/2; i++)
#define LOOP_XYZ(i, j, k, ix, iy, iz) \
        for (i = 0; i <= ix; i++) \
        for (j = 0; j <= iy; j++) \
        for (k = 0; k <= iz; k++)

static int _offset_cart[] = {0, 1, 4, 10, 20, 35, 56, 84, 120,
                             165, 220, 286, 364, 455, 560};

/*
 * Matrix of angular moment operator l*1j on the real spherical harmonic basis.
 * Obtained by the function below for l = 0..4
 *
def angular_moment_matrix(l):
    lz = numpy.diag(numpy.arange(-l, l+1, dtype=numpy.complex))
    lx = numpy.zeros_like(lz)
    ly = numpy.zeros_like(lz)
    for mi in range(-l, l+1):
        mj = mi + 1
        if mj <= l:
            lx[l+mi,l+mj] = .5  * ((l+mj)*(l-mj+1))**.5
            ly[l+mi,l+mj] = .5j * ((l+mj)*(l-mj+1))**.5
        mj = mi - 1
        if mj >= -l:
            lx[l+mi,l+mj] = .5  * ((l-mj)*(l+mj+1))**.5
            ly[l+mi,l+mj] =-.5j * ((l-mj)*(l+mj+1))**.5
    u = sph_pure2real(l)
    lx = u.conj().T.dot(lx).dot(u)
    ly = u.conj().T.dot(ly).dot(u)
    lz = u.conj().T.dot(lz).dot(u)
    return numpy.array((lx, ly, lz)) * 1j
*/
static double _angular_moment_matrix_s[] = {
        0, 0, 0
};
static double _angular_moment_matrix_p[] = {
        0, 0, 0, 0, 0, 1, 0, -1, 0,
        0, 0, -1, 0, 0, 0, 1, 0, 0,
        0, 1, 0, -1, 0, 0, 0, 0, 0,
};
static double _angular_moment_matrix_d[] = {
        0, 0, 0, 1, 0, 0, 0, 1.73205080756887719, 0, 1, 0, -1.73205080756887719, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0,
        0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1.73205080756887719, 0, 0, 0, -1.73205080756887719, 0, 1, 0, 0, 0, -1, 0,
        0, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0,
};
static double _angular_moment_matrix_f[] = {
        0, 0, 0, 0, 0, 1.22474487139158894, 0, 0, 0, 0, 0, 1.58113883008418976, 0, 1.22474487139158894, 0, 0, 0, 2.44948974278317788, 0, 1.58113883008418976, 0, 0, 0, -2.44948974278317788, 0, 0, 0, 0, 0, -1.58113883008418976, 0, 0, 0, 0, 0, -1.22474487139158894, 0, -1.58113883008418976, 0, 0, 0, 0, 0, -1.22474487139158894, 0, 0, 0, 0, 0,
        0, -1.22474487139158894, 0, 0, 0, 0, 0, 1.22474487139158894, 0, -1.58113883008418976, 0, 0, 0, 0, 0, 1.58113883008418976, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2.44948974278317788, 0, 0, 0, 0, 0, -2.44948974278317788, 0, 1.58113883008418976, 0, 0, 0, 0, 0, -1.58113883008418976, 0, 1.22474487139158894, 0, 0, 0, 0, 0, -1.22474487139158894, 0,
        0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0,
};
static double _angular_moment_matrix_g[] = {
  0, 0, 0, 0, 0, 0, 0, 1.41421356237309515, 0, 0, 0, 0, 0, 0, 0, 1.87082869338697066, 0, 1.41421356237309515, 0, 0, 0, 0, 0, 2.12132034355964239, 0, 1.87082869338697066, 0, 0, 0, 0, 0, 3.16227766016837952, 0, 2.12132034355964239, 0, 0, 0, 0, 0, -3.16227766016837952, 0, 0, 0, 0, 0, 0, 0, -2.12132034355964239, 0, 0, 0, 0, 0, 0, 0, -1.87082869338697066, 0, -2.12132034355964239, 0, 0, 0, 0, 0, -1.41421356237309515, 0, -1.87082869338697066, 0, 0, 0, 0, 0, 0, 0, -1.41421356237309515, 0, 0, 0, 0, 0, 0, 0,
  0, -1.41421356237309515, 0, 0, 0, 0, 0, 0, 0, 1.41421356237309515, 0, -1.87082869338697066, 0, 0, 0, 0, 0, 0, 0, 1.87082869338697066, 0, -2.12132034355964239, 0, 0, 0, 0, 0, 0, 0, 2.12132034355964239, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3.16227766016837952, 0, 0, 0, 0, 0, 0, 0, -3.16227766016837952, 0, 2.12132034355964239, 0, 0, 0, 0, 0, 0, 0, -2.12132034355964239, 0, 1.87082869338697066, 0, 0, 0, 0, 0, 0, 0, -1.87082869338697066, 0, 1.41421356237309515, 0, 0, 0, 0, 0, 0, 0, -1.41421356237309515, 0,
  0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0,
};
static double *_angular_moment_matrix[] = {
        _angular_moment_matrix_s,
        _angular_moment_matrix_p,
        _angular_moment_matrix_d,
        _angular_moment_matrix_f,
        _angular_moment_matrix_g,
};
// einsum('jfnq,lmn->ljfmq', angj, jmm)
static void transform_angj(double *jmm_angj, double *angj, int lj, int lc)
{
        const int lj1 = lj + 1;
        const int dlc = lc * 2 + 1;
        const int ljlc1 = lj + lc + 1;
        const int nfj = (lj+1) * (lj+2) / 2;
        double *jmm;

        const double D0 = 0;
        const double D1 = 1;
        const char TRANS_N = 'N';

        int i, j;
        for (i = 0; i < 3; i++) {
                jmm = _angular_moment_matrix[lc] + i*dlc*dlc;
                for (j = 0; j < lj1*nfj; j++) {
                        dgemm_(&TRANS_N, &TRANS_N, &ljlc1, &dlc, &dlc,
                               &D1, angj+j*dlc*ljlc1, &ljlc1, jmm, &dlc,
                               &D0, jmm_angj+j*dlc*ljlc1, &ljlc1);
                }
                jmm_angj += lj1 * nfj * dlc * ljlc1;
        }
}

/*
 * exponentially scaled modified spherical Bessel function of the first kind
 * scipy.special.sph_in(order, z) * numpy.exp(-z)
 *
 * JCC, 27, 1009
 */
void ECPsph_ine(double *out, int order, double z)
{
        int i, k;
        if (z < 1e-7) {
                // (1-z) * z^l / (2l+1)!!
                out[0] = 1. - z;
                for (i = 1; i <= order; i++) {
                        out[i] = out[i-1] * z / (i*2+1);
                }
        } else if (z > 16) {
                // R_l(z) = \sum_k (l+k)!/(k!(l-k)!(2x)^k)
                double z2 = -.5 / z;
                double ti, s;
                for (i = 0; i <= order; i++) {
                        ti = .5 / z;
                        s = ti;
                        for (k = 1; k <= i; k++) {
                                ti *= z2;
                                s += ti * _factorial[i+k]
                                        / (_factorial[k] * _factorial[i-k]);
                        }
                        out[i] = s;
                }
        } else {
                // z^l e^{-z} \sum (z^2/2)^k/(k!(2k+2l+1)!!)
                double z2 = .5 * z * z;
                double t0 = exp(-z);
                double ti, s, next;
                for (i = 0; i <= order; i++) {
                        ti = t0;
                        s = ti;
                        for (k = 1;; k++) {
                                ti *= z2 / (k * (k*2+i*2+1));
                                next = s + ti;
                                if (next == s) {
                                        break;
                                } else {
                                        s = next;
                                }
                        }
                        t0 *= z/(i*2+3);  // k = 0
                        out[i] = s;
                }
        }

}


static const double _l2[] = { // l/(2l+1)
        0., 1./3, 2./5, 3./7, 4./9, 5./11, 6./13, 7./15,
        8./17, 9./19, 10./21, 11./23, 12./25, 13./27,
        14./29, 15./31, 16./33, 17./35, 18./37, 19./39,
        20./41, 21./43, 22./45, 23./47,
};
static const double _j_inv[] = { // 1/j
        0., 1., .5, 1./3, 1./4, 1./5, 1./6, 1./7, 1./8, 1./9
};
void ECPsph_ine_opt(double *out, int order, double z)
{
        if (z < 1e-7 || z > 16) {
                return ECPsph_ine(out, order, z);
        } else {
                /*
                 * j_n' = j_{n-1} - (n+1)/z j_n = j_{n+1} + n/z j_n
                 *      => n/(2n+1) j_{n-1} + (n+1)/(2n+1) j_{n+1}
                 * K_n' = n/(2n+1) K_{n-1} + (n+1)/(2n+1) K_{n+1} - K_n
                 */
                int i, j;
                double buf[K_TAB_COL*2];
                double *k0 = buf;
                double *k1 = buf + K_TAB_COL;
                double *tmp;
                int entry = (int)floor(z/K_TAB_INTERVAL);
                double *tabu = _sph_ine_tab_order7 + entry * ORDER7OFFSET * (K_TAYLOR_MAX+1);
                double z0 = entry*K_TAB_INTERVAL + K_TAB_INTERVAL/2;
                double dz = z - z0;
                double fac = 1;
                double s[ORDER7OFFSET];

                switch (order) {
                case 0:
                        s[0] = tabu[0];
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                s[0] += tabu[j*ORDER7OFFSET] * fac;
                        }
                        out[0] = s[0];
                        break;
                case 1:
                        s[0] = tabu[0];
                        s[1] = tabu[1];
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                s[0] += tabu[j*ORDER7OFFSET+0] * fac;
                                s[1] += tabu[j*ORDER7OFFSET+1] * fac;
                        }
                        out[0] = s[0];
                        out[1] = s[1];
                        break;
                case 2:
                        s[0] = tabu[0];
                        s[1] = tabu[1];
                        s[2] = tabu[2];
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                s[0] += tabu[j*ORDER7OFFSET+0] * fac;
                                s[1] += tabu[j*ORDER7OFFSET+1] * fac;
                                s[2] += tabu[j*ORDER7OFFSET+2] * fac;
                        }
                        out[0] = s[0];
                        out[1] = s[1];
                        out[2] = s[2];
                        break;
                case 3:
                        for (i = 0; i <= 3; i++) {
                                s[i] = tabu[i];
                        }
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= 3; i++) {
                                        s[i] += tabu[j*ORDER7OFFSET+i] * fac;
                                }
                        }
                        for (i = 0; i <= 3; i++) {
                                out[i] = s[i];
                        }
                        break;
                case 4:
                        for (i = 0; i <= 4; i++) {
                                s[i] = tabu[i];
                        }
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= 4; i++) {
                                        s[i] += tabu[j*ORDER7OFFSET+i] * fac;
                                }
                        }
                        for (i = 0; i <= 4; i++) {
                                out[i] = s[i];
                        }
                        break;
                case 5:
                        for (i = 0; i <= 5; i++) {
                                s[i] = tabu[i];
                        }
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= 5; i++) {
                                        s[i] += tabu[j*ORDER7OFFSET+i] * fac;
                                }
                        }
                        for (i = 0; i <= 5; i++) {
                                out[i] = s[i];
                        }
                        break;
                case 6:
                        for (i = 0; i <= 6; i++) {
                                s[i] = tabu[i];
                        }
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= 6; i++) {
                                        s[i] += tabu[j*ORDER7OFFSET+i] * fac;
                                }
                        }
                        for (i = 0; i <= 6; i++) {
                                out[i] = s[i];
                        }
                        break;
                case 7:
                        for (i = 0; i <= 7; i++) {
                                s[i] = tabu[i];
                        }
                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= 7; i++) {
                                        s[i] += tabu[j*ORDER7OFFSET+i] * fac;
                                }
                        }
                        for (i = 0; i <= 7; i++) {
                                out[i] = s[i];
                        }
                        break;
                default:
                        tabu = _sph_ine_tab + entry * K_TAB_COL;
                        for (i = 0; i <= order+K_TAYLOR_MAX; i++) {
                                k0[i] = tabu[i];
                        }
                        for (i = 0; i <= order; i++) {
                                out[i] = k0[i];
                        }

                        for (j = 1; j <= K_TAYLOR_MAX; j++) {
                                k1[0] = k0[1] - k0[0];
                                for (i = 1; i <= order+K_TAYLOR_MAX-j; i++) {
                                        k1[i] = _l2[i]*k0[i-1] + (1-_l2[i])*k0[i+1] - k0[i];
                                }
                                fac *= dz * _j_inv[j];
                                for (i = 0; i <= order; i++) {
                                        out[i] += k1[i] * fac;
                                }
                                tmp = k0;
                                k0 = k1;
                                k1 = tmp;
                        }
                }
        }
}

void ECPsph_ine_a(double *out, int order, double *zs, int n)
{
        int i;
        for (i = 0; i < n; i++) {
                ECPsph_ine(out, order, zs[i]);
                out += order+1;
        }
}

void ECPgauss_chebyshev(double *rs, double *ws, int n)
{
        int i;
        double step = 1./(n+1);
        double fac = 16 * step / 3;
        double xinc = M_PI * step;
        double x1 = 0;
        double x2, x3, x4, xi;
        for (i = 0; i < n; i++) {
                x1 += xinc;
                x2 = sin(x1);
                x3 = sin(x1*2);
                x4 = x2 * x2;
                xi = (n-i*2-1) * step + M_1_PI * (1+2./3.*x4) * x3;
                rs[i] = 1 - log(1+xi) * M_LOG2E;  // 1/ln2
                ws[i] = fac * x4 * x4 * M_LOG2E / (1+xi);
        }
}

/*
 * Return the number of effective grids
 */
int ECPrad_part(double *ur, double *rs, int rs_off, int nrs, int inc,
                int *ecpshls, int *ecpbas,
                int *atm, int natm, int *bas, int nbas, double *env, ECPOpt *opt, double *cache)
{
        const int ecpsh0 = ecpshls[0];
        const int ecpsh1 = ecpshls[1];
        double *ubuf, *r2;
        MALLOC_INSTACK(ubuf, nrs);
        MALLOC_INSTACK(r2, nrs);
        double *ak, *ck, *u_ecp;
        double s;
        int npk;
        int ish, i, kp, n;
        int nrs_now;
        int nrs_max = 0;

        rs += rs_off;
        for (i = 0; i < nrs; i++) {
                r2[i] = rs[i*inc] * rs[i*inc];
                ur[i] = 0;
        }

        for (ish = ecpsh0; ish < ecpsh1; ish++) {
                if (opt == NULL) {
                        npk = ecpbas[ish*BAS_SLOTS+NPRIM_OF];
                        ak = env + ecpbas[ish*BAS_SLOTS+PTR_EXP];
                        ck = env + ecpbas[ish*BAS_SLOTS+PTR_COEFF];

                        for (i = 0; i < nrs; i++) {
                                s = ck[0] * exp(-ak[0]*r2[i]);
                                for (kp = 1; kp < npk; kp++) {
                                        s += ck[kp] * exp(-ak[kp]*r2[i]);
                                }
                                ubuf[i] = s;
                                if (i > 2 &&
                                    fabs(ubuf[i]) < SIM_ZERO &&
                                    fabs(ubuf[i-1]) < SIM_ZERO) {
                                        break;
                                }
                        }
                } else {
                        u_ecp = opt->u_ecp + ish * (1 << LEVEL_MAX);
                        for (i = 0; i < nrs; i++) {
                                ubuf[i] = u_ecp[i*inc+rs_off];
                                if (fabs(ubuf[i]) < SIM_ZERO) {
                                        break;
                                }
                        }
                }
                nrs_now = i;
                nrs_max = MAX(nrs_max, nrs_now);
                switch (ecpbas[ish*BAS_SLOTS+RADI_POWER]) {
                case 1:
                        for (i = 0; i < nrs_now; i++) {
                                ubuf[i] *= rs[i*inc];
                        }
                        break;
                case 2:
                        for (i = 0; i < nrs_now; i++) {
                                ubuf[i] *= r2[i];
                        }
                        break;
                case 3:
                        for (i = 0; i < nrs_now; i++) {
                                ubuf[i] *= r2[i] * rs[i*inc];
                        }
                        break;
                default:
                        for (i = 0; i < nrs_now; i++) {
                        for (n = 0; n < ecpbas[ish*BAS_SLOTS+RADI_POWER]; n++) {
                                ubuf[i] *= rs[i*inc];
                        } }
                        break;
                }

                for (i = 0; i < nrs_now; i++) {
                        ur[i] += ubuf[i];
                }
        }
        return nrs_max;
}

static double int_unit_xyz(int i, int j, int k)
{
        if (i % 2 || j % 2 || k % 2) {
                return 0;
        } else {
                return (factorial2(i-1) * factorial2(j-1) *
                        factorial2(k-1) / factorial2(i+j+k+1));
        }
}
/*
 * Angular part integration then transform back to cartesian basis
 */
double *CINTc2s_bra_sph(double *gsph, int nket, double *gcart, int l);
double *CINTs2c_bra_sph(double *gsph, int nket, double *gcart, int l);
static void ang_nuc_in_cart(double *omega, int l, double *r)
{
        double buf[CART_MAX];
        double xx[16];
        double yy[16];
        double zz[16];
        int i, j, k, n;

        switch (l) {
        case 0:
                omega[0] = 0.07957747154594767;
                break;
        case 1:
                omega[0] = r[0] * 0.2387324146378430;
                omega[1] = r[1] * 0.2387324146378430;
                omega[2] = r[2] * 0.2387324146378430;
                break;
        default:
                xx[0] = 1;
                yy[0] = 1;
                zz[0] = 1;
                for (i = 1; i <= l; i++) {
                        xx[i] = xx[i-1] * r[0];
                        yy[i] = yy[i-1] * r[1];
                        zz[i] = zz[i-1] * r[2];
                }
                for (n = 0, i = l; i >= 0; i--) {
                        for (j = l-i; j >= 0; j--, n++) {
                                k = l - i - j;
                                omega[n] = xx[i] * yy[j] * zz[k];
                        }
                }
                CINTc2s_bra_sph(buf, 1, omega, l);
                CINTs2c_bra_sph(buf, 1, omega, l);
        }
}

static void cache_3dfac(double *facs, int l, double *r)
{
        int l1 = l + 1;
        double *facx = facs;
        double *facy = facs + l1*l1;
        double *facz = facy + l1*l1;
        double xx[16];
        double yy[16];
        double zz[16];
        double bfac;
        int i, j, off;
        xx[0] = 1;
        yy[0] = 1;
        zz[0] = 1;
        for (i = 1; i <= l; i++) {
                xx[i] = xx[i-1] * r[0];
                yy[i] = yy[i-1] * r[1];
                zz[i] = zz[i-1] * r[2];
        }
        for (i = 0; i <= l; i++) {
                for (j = 0; j <= i; j++) {
                        bfac = binom(i,j);
                        off = i*l1+j;
                        facx[off] = bfac * xx[i-j];
                        facy[off] = bfac * yy[i-j];
                        facz[off] = bfac * zz[i-j];
                }
        }
}

void type2_facs_ang(double *facs, int li, int lc, double *ri, double *cache)
{
        double unitr[3];
        if (ri[0] == 0 && ri[1] == 0 && ri[2] == 0) {
                unitr[0] = 0;
                unitr[1] = 0;
                unitr[2] = 0;
        } else {
                double norm_ri = -1/sqrt(SQUARE(ri));
                unitr[0] = ri[0] * norm_ri;
                unitr[1] = ri[1] * norm_ri;
                unitr[2] = ri[2] * norm_ri;
        }

        const int li1 = li + 1;
        const int dlc = lc * 2 + 1;
        const int dlambda = li + lc + 1;
        double omega_nuc[CART_CUM];
        double *pnuc;
        int m, n, i, j, k, lmb, mi;
        for (i = 0; i <= li+lc; i++) {
                pnuc = omega_nuc + _offset_cart[i];
                ang_nuc_in_cart(pnuc, i, unitr);
        }
        for (i = 0; i < _offset_cart[li+lc+1]; i++) {
                omega_nuc[i] *= 4 * M_PI;
        }

        double buf[CART_MAX];
        double *omega;
        MALLOC_INSTACK(omega, li1*li1*li1*dlambda*dlc);
        double *pomega;
        int dlclmb = dlambda * dlc;
        int need_even;
        int need_odd;
        int pu, pv, pw, pr, ps, pt;
        for (i = 0; i <= li; i++) {
        for (j = 0; j <= li-i; j++) {
        for (k = 0; k <= li-i-j; k++) {
                // use need_even to ensure (lc+a+b+c+lmb) is even
                need_even = (lc+i+j+k)%2;
                pomega = omega + (i*li1*li1+j*li1+k)*dlclmb+need_even*dlc;
                for (lmb = need_even; lmb <= li+lc; lmb+=2) {
                        pnuc = omega_nuc + _offset_cart[lmb];
                        LOOP_CART(lc, m, pu, pv, pw) {
                                buf[m] = 0;
                                LOOP_CART(lmb, n, pr, ps, pt) {
                                        buf[m] += pnuc[n] *
                                                int_unit_xyz(i+pu+pr, j+pv+ps, k+pw+pt);
                                }
                        }
                        switch (lc) {
                        case 0:
                                pomega[0] = buf[0] * 0.282094791773878143;
                                break;
                        case 1:
                                pomega[0] = buf[0] * 0.488602511902919921;
                                pomega[1] = buf[1] * 0.488602511902919921;
                                pomega[2] = buf[2] * 0.488602511902919921;
                                break;
                        default:
                                CINTc2s_bra_sph(pomega, 1, buf, lc);
                        }
                        pomega += dlc*2;
                }

                need_odd = need_even ^ 1;
                pomega = omega + (i*li1*li1+j*li1+k)*dlclmb+need_odd*dlc;
                for (lmb = need_odd; lmb <= li+lc; lmb+=2) {
                        for (m = 0; m < dlc; m++) {
                                pomega[m] = 0;
                        }
                        pomega += dlc*2;
                }
        } } }

        const int nfi = _offset_cart[li+1]-_offset_cart[li];  //(li+1)*(li+2)/2;
        double *fac3d;
        MALLOC_INSTACK(fac3d, 3*li1*li1);
        double *fac3dx = fac3d;
        double *fac3dy = fac3dx + li1*li1;
        double *fac3dz = fac3dy + li1*li1;
        double *pfacs;
        double fac;
        cache_3dfac(fac3d, li, ri);

        for (i = 0; i < li1*nfi*dlclmb; i++) { facs[i] = 0; }
        LOOP_CART(li, mi, pr, ps, pt) {
                LOOP_XYZ(i, j, k, pr, ps, pt) {
                        need_even = (lc+i+j+k)%2;
                        fac = fac3dx[pr*li1+i] * fac3dy[ps*li1+j] * fac3dz[pt*li1+k];
                        pomega = omega + (i*li1*li1+j*li1+k)*dlclmb;
                        pfacs = facs + ((i+j+k)*nfi+mi)*dlclmb;
                        for (m = 0; m < dlc; m++) {
                        for (n = need_even; n < dlambda; n+=2) {
                                pfacs[m*dlambda+n] += fac * pomega[n*dlc+m];
                        } }
                }
        }
}

void type2_facs_rad(double *facs, int ish, int lc, double rca,
                    double *rs, int nrs, int inc,
                    int *atm, int natm, int *bas, int nbas, double *env, double *cache)
{
        if (nrs == 0) {
                return;
        }

        const int li = bas[ANG_OF  +ish*BAS_SLOTS];
        const int np = bas[NPRIM_OF+ish*BAS_SLOTS];
        const int nc = bas[NCTR_OF +ish*BAS_SLOTS];
        const double *ai = env + bas[PTR_EXP+ish*BAS_SLOTS];
        const double *ci = env + bas[PTR_COEFF+ish*BAS_SLOTS];
        const int lilc1 = li + lc + 1;
        int ip, i, j;
        double ka;
        double *r2, *buf;
        MALLOC_INSTACK(r2, nrs);
        MALLOC_INSTACK(buf, np*nrs*lilc1);
        double t1, ar2;
        double *pbuf = buf;

        for (i = 0; i < nrs; i++) {
                t1 = rs[i*inc] - rca;
                r2[i] = t1 * t1;
        }
        for (ip = 0; ip < np; ip++) {
                ka = 2. * ai[ip] * rca;
                for (i = 0; i < nrs; i++) {
                        // + 6. to include the largest value in ECPsph_ine_opt
                        ar2 = ai[ip] * r2[i];
                        if (ar2 > EXPCUTOFF + 6.) {
                                for (j = 0; j <= li+lc; j++) {
                                        pbuf[j] = 0;
                                }
                        } else {
                                t1 = exp(-ar2);
                                ECPsph_ine_opt(pbuf, li+lc, ka*rs[i*inc]);
                                for (j = 0; j <= li+lc; j++) {
                                        pbuf[j] *= t1;
                                }
                        }
                        pbuf += lilc1;
                }
        }

        const double D0 = 0;
        const double D1 = 1;
        const char TRANS_N = 'N';
        const int m = nrs * lilc1;
        dgemm_(&TRANS_N, &TRANS_N, &m, &nc, &np,
               &D1, buf, &m, ci, &np, &D0, facs, &m);
}

void type1_static_facs(double *facs, int li, double *ri, double *cache)
{
        const int d1 = li + 1;
        const int d2 = d1 * d1;
        const int d3 = d2 * d1;
        double *fac3d;
        MALLOC_INSTACK(fac3d, 3*d1*d1);
        double *fac3dx = fac3d;
        double *fac3dy = fac3dx + d1*d1;
        double *fac3dz = fac3dy + d1*d1;
        double *pfacs;
        cache_3dfac(fac3d, li, ri);
        int mi, i, j, k, px, py, pz;

        LOOP_CART(li, mi, px, py, pz) {
                pfacs = facs + mi * d3;
                LOOP_XYZ(i, j, k, px, py, pz) {
                        pfacs[i*d2+j*d1+k] = fac3dx[px*d1+i] * fac3dy[py*d1+j]
                                           * fac3dz[pz*d1+k];
                }
        }
}

void type1_rad_ang(double *rad_ang, int lmax, double *r, double *rad_all)
{
        double unitr[3];
        if (r[0] == 0 && r[1] == 0 && r[2] == 0) {
                unitr[0] = 0;
                unitr[1] = 0;
                unitr[2] = 0;
        } else {
                double norm_r = -1/sqrt(SQUARE(r));
                unitr[0] = r[0] * norm_r;
                unitr[1] = r[1] * norm_r;
                unitr[2] = r[2] * norm_r;
        }

        double omega_nuc[CART_CUM];
        double *pnuc;
        int n, i, j, k, lmb;
        for (i = 0; i <= lmax; i++) {
                pnuc = omega_nuc + _offset_cart[i];
                ang_nuc_in_cart(pnuc, i, unitr);
        }

        const int d1 = lmax + 1;
        const int d2 = d1 * d1;
        const int d3 = d2 * d1;
        int need_even;
        double tmp;
        int pr, ps, pt;
        double *pout, *prad;
        for (i = 0; i < d3; i++) { rad_ang[i] = 0; }
        for (i = 0; i <= lmax; i++) {
        for (j = 0; j <= lmax-i; j++) {
        for (k = 0; k <= lmax-i-j; k++) {
                pout = rad_ang + i*d2+j*d1+k;
                prad = rad_all + (i+j+k)*d1;
                // need_even to ensure (a+b+c+lmb) is even
                need_even = (i+j+k)%2;
                for (lmb = need_even; lmb <= lmax; lmb+=2) {
                        tmp = 0;
                        pnuc = omega_nuc + _offset_cart[lmb];
                        LOOP_CART(lmb, n, pr, ps, pt) {
                                tmp += pnuc[n] * int_unit_xyz(i+pr, j+ps, k+pt);
                        }
                        *pout += prad[lmb] * tmp;
                }
        } } }
}

// ecpbasis which have different r^n but the same l can be put in the same shell
// and evaluated together in ECPrad_part function
static int _loc_ecpbas(int *ecploc, int *ecpbas, int necpbas)
{
        int i, l, so, atm_id;
        ecploc[0] = 0;
        int nslots = 0;
        int atm_last = ecpbas[ATOM_OF];
        int l_last = ecpbas[ANG_OF];
        int so_last = ecpbas[SO_TYPE_OF];
        for (i = 1; i < necpbas; i++) {
                atm_id = ecpbas[ATOM_OF+i*BAS_SLOTS];
                l = ecpbas[ANG_OF +i*BAS_SLOTS];
                so = ecpbas[SO_TYPE_OF + i*BAS_SLOTS];
                if (atm_id != atm_last || l != l_last || so != so_last) {
                        nslots++;
                        ecploc[nslots] = i;
                        atm_last = atm_id;
                        l_last = l;
                        so_last = so;
                }
        }
        if (necpbas != 0) {
                nslots++;
                ecploc[nslots] = necpbas;  // store boundary at last
        }
        return nslots;
}

static double distance_square(const double *r1, const double *r2)
{
        double r12[3];
        r12[0] = r1[0] - r2[0];
        r12[1] = r1[1] - r2[1];
        r12[2] = r1[2] - r2[2];
        return SQUARE(r12);
}

static int check_3c_overlap(int *shls, int *atm, int *bas, double *env,
                            double *rc, int *ecpshls, int *ecpbas)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int ecpsh0 = ecpshls[0];
        const int ecpsh1 = ecpshls[1];
        const int iprim = bas[NPRIM_OF+ish*BAS_SLOTS];
        const int jprim = bas[NPRIM_OF+jsh*BAS_SLOTS];
        const double ai = env[bas[PTR_EXP+ish*BAS_SLOTS]+iprim-1];
        const double aj = env[bas[PTR_EXP+jsh*BAS_SLOTS]+jprim-1];
        const double *ri = env + atm[PTR_COORD+bas[ATOM_OF+ish*BAS_SLOTS]*ATM_SLOTS];
        const double *rj = env + atm[PTR_COORD+bas[ATOM_OF+jsh*BAS_SLOTS]*ATM_SLOTS];
        double rrab = distance_square(ri, rj);
        double rrca = distance_square(rc, ri);
        double rrcb = distance_square(rc, rj);
        double aiaj = ai * aj * rrab;
        double aij = ai + aj;
        double ajak, aiak, aijk, eijk;
        double *ak;
        int csh, kprim;

        for (csh = ecpsh0; csh < ecpsh1; csh++) {
                kprim = ecpbas[csh*BAS_SLOTS+NPRIM_OF];
                ak = env + ecpbas[csh*BAS_SLOTS+PTR_EXP];

                // Test the last primitive funciton only because the basis
                // functions are sorted so that the last one has the smallest
                // exponent.
                aijk = aij + ak[kprim-1];
                aiak = ai * ak[kprim-1] * rrca;
                ajak = aj * ak[kprim-1] * rrcb;
                eijk = (aiaj+aiak+ajak) / aijk;
                if (eijk < EXPCUTOFF) {
                        return 1;
                }
        }
        return 0;
}

int ECPtype2_cart(double *gctr, int *shls, int *ecpbas, int necpbas,
                  int *atm, int natm, int *bas, int nbas, double *env,
                  ECPOpt *opt, double *cache)
{
        if (necpbas == 0) {
                return 0;
        }
        double *cache0 = cache;
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int di = nfi * nci;
        const int ngctr = nci * ncj * nfi * nfj;
        const double *ri = env + atm[PTR_COORD+bas[ATOM_OF+ish*BAS_SLOTS]*ATM_SLOTS];
        const double *rj = env + atm[PTR_COORD+bas[ATOM_OF+jsh*BAS_SLOTS]*ATM_SLOTS];
        const double D0 = 0;
        const double D1 = 1;
        const char TRANS_N = 'N';
        const char TRANS_T = 'T';
        const double common_fac = CINTcommon_fac_sp(li) *
                                  CINTcommon_fac_sp(lj) * 16 * M_PI * M_PI;

        int nrs = 1 << LEVEL_MAX;
        int atm_id, lc, lab, dlc, im, mq;
        int i, j, n, ic, jc;
        double dca, dcb, s;
        double *rc, *pradi, *pradj, *prur;
        double rca[3];
        double rcb[3];
        int has_value = 0;
        int *ecploc;
        MALLOC_INSTACK(ecploc, necpbas+1);
        int nslots = _loc_ecpbas(ecploc, ecpbas, necpbas);
        int *ecpshls;
        int iloc;

        double *rur, *radi, *radj, *angi, *angj, *rad_all, *buf;
        double *prad, *plast;
        MALLOC_INSTACK(rad_all, nci*ncj*(li+lj+1)*(li+ECP_LMAX+1)*(lj+ECP_LMAX+1));
        MARK_STACK;
        MALLOC_INSTACK(angi, (li+1)*nfi*(ECP_LMAX*2+1)*(li+ECP_LMAX+1));
        MALLOC_INSTACK(angj, (lj+1)*nfj*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));
        MALLOC_INSTACK(buf, nfi*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));

        RESTORE_STACK;
        MALLOC_INSTACK(rur, nrs*(li+lj+1));
        MALLOC_INSTACK(radi, nci*(li+ECP_LMAX+1)*nrs);
        MALLOC_INSTACK(radj, ncj*(lj+ECP_LMAX+1)*nrs);
        int lilj1 = li + lj + 1;
        int lilc1 = li + ECP_LMAX + 1;
        int ljlc1 = lj + ECP_LMAX + 1;
        int d2 = lilc1 * ljlc1;
        int d3 = lilj1 * d2;
        MALLOC_INSTACK(plast, d2);
        char *converged;
        MALLOC_INSTACK(converged, nci*ncj*lilj1);

        for (i = 0; i < ngctr; i++) { gctr[i] = 0; }

        for (iloc = 0; iloc < nslots; iloc++) {
                lc = ecpbas[ANG_OF+ecploc[iloc]*BAS_SLOTS];
                if (lc == -1 ||
                    ecpbas[SO_TYPE_OF+ecploc[iloc]*BAS_SLOTS] == 1) {
                        continue;
                }
                atm_id = ecpbas[ATOM_OF+ecploc[iloc]*BAS_SLOTS];
                rc = env + atm[PTR_COORD+atm_id*ATM_SLOTS];
                ecpshls = ecploc + iloc;

        if (!check_3c_overlap(shls, atm, bas, env, rc, ecpshls, ecpbas)) {
                continue;
        }
        has_value = 1;
        rca[0] = rc[0] - ri[0];
        rca[1] = rc[1] - ri[1];
        rca[2] = rc[2] - ri[2];
        rcb[0] = rc[0] - rj[0];
        rcb[1] = rc[1] - rj[1];
        rcb[2] = rc[2] - rj[2];
        dca = sqrt(SQUARE(rca));
        dcb = sqrt(SQUARE(rcb));

        dlc = lc * 2 + 1;
        lilc1 = li + lc + 1;
        ljlc1 = lj + lc + 1;
        d2 = lilc1 * ljlc1;
        d3 = lilj1 * d2;
        im = nfi * dlc;
        mq = dlc * ljlc1;

        int level, nrs0, start, step, ijl;
        double wtscale;
        char all_conv;
        double *rs = rs_gauss_chebyshev2047;
        double *ws = ws_gauss_chebyshev2047;
        for (i = 0; i < nci*ncj*lilj1; i++) { converged[i] = 0; }
        for (i = 0; i < nci*ncj*d3; i++) { rad_all[i] = 0; }
        nrs0 = (1 << LEVEL0) - 1;
        step = 1 << (LEVEL_MAX - LEVEL0);
        start = step - 1;
        wtscale = step;
        for (level = LEVEL0; level <= LEVEL_MAX; level++) {
                nrs = ECPrad_part(rur, rs, start, nrs0, step, ecpshls, ecpbas,
                                  atm, natm, bas, nbas, env, opt, cache);
                for (i = 0; i < nrs; i++) {
                        rur[i] *= ws[start+i*step] * wtscale;
                        for (lab = 1; lab <= li+lj; lab++) {
                                rur[nrs*lab+i] = rur[nrs*(lab-1)+i] * rs[start+i*step];
                        }
                }

                type2_facs_rad(radi, ish, lc, dca, rs+start, nrs, step,
                               atm, natm, bas, nbas, env, cache);
                type2_facs_rad(radj, jsh, lc, dcb, rs+start, nrs, step,
                               atm, natm, bas, nbas, env, cache);
                all_conv = 1;
                for (ijl = 0, ic = 0; ic < nci; ic++) {
                for (jc = 0; jc < ncj; jc++) {
                        pradi = radi + ic * nrs * lilc1;
                        pradj = radj + jc * nrs * ljlc1;
                        for (lab = 0; lab <= li+lj; lab++, ijl++) {
                                if (!converged[ijl]) {
        prur = rur + lab * nrs;
        prad = rad_all + ijl*d2;
        for (i = 0; i < d2; i++) {
                plast[i] = prad[i];
                prad[i] *= .5;
        }

        for (i = 0; i < lilc1; i++) {
        for (j = 0; j < ljlc1; j++) {
                s = prad[i*ljlc1+j];
                for (n = 0; n < nrs; n++) {
                        s += prur[n] * pradi[n*lilc1+i] * pradj[n*ljlc1+j];
                }
                prad[i*ljlc1+j] = s;
        } }

        for (i = 0; i < d2; i++) {
                if (!CLOSE_ENOUGH(plast[i],prad[i])) {
                        converged[ijl] = 0;
                        all_conv = 0;
                        break;
                }
        }
                                }
                        }
                } }

                if (all_conv) {
                        break;
                } else {
                        nrs0 = (1 << level) - 1;
                        step = 1 << (LEVEL_MAX - level);
                        start = (start - 1) / 2;
                        wtscale *= .5;
                }
        }

        type2_facs_ang(angi, li, lc, rca, cache);
        type2_facs_ang(angj, lj, lc, rcb, cache);
        for (ic = 0; ic < nci; ic++) {
        for (jc = 0; jc < ncj; jc++) {
                prad = rad_all + (ic*ncj+jc)*d3;
                for (i = 0; i <= li; i++) {
                for (j = 0; j <= lj; j++) {
                        dgemm_(&TRANS_N, &TRANS_N, &ljlc1, &im, &lilc1,
                               &D1, prad+(i+j)*d2, &ljlc1,
                               angi+i*nfi*dlc*lilc1, &lilc1, &D0, buf, &ljlc1);
                        dgemm_(&TRANS_T, &TRANS_N, &nfi, &nfj, &mq,
                               &common_fac, buf, &mq, angj+j*nfj*dlc*ljlc1, &mq,
                               &D1, gctr+jc*nfj*di+ic*nfi, &di);
                } }
        } }
        }
        return has_value;
}

int ECPtype_so_cart(double *gctr, int *shls, int *ecpbas, int necpbas,
                    int *atm, int natm, int *bas, int nbas, double *env,
                    ECPOpt *opt, double *cache)
{
        if (necpbas == 0) {
                return 0;
        }
        double *cache0 = cache;
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int di = nfi * nci;
        const int ngctr = nci * ncj * nfi * nfj;
        const double *ri = env + atm[PTR_COORD+bas[ATOM_OF+ish*BAS_SLOTS]*ATM_SLOTS];
        const double *rj = env + atm[PTR_COORD+bas[ATOM_OF+jsh*BAS_SLOTS]*ATM_SLOTS];

        int nrs = 1 << LEVEL_MAX;
        int ecploc[necpbas+1];
        int nslots = _loc_ecpbas(ecploc, ecpbas, necpbas);
        int *ecpshls;
        int iloc;

        const double D0 = 0;
        const double D1 = 1;
        const char TRANS_N = 'N';
        const char TRANS_T = 'T';
        double common_fac = CINTcommon_fac_sp(li) *
                            CINTcommon_fac_sp(lj) * 16 * M_PI * M_PI;
        // be careful with the factor 1/2 for the spin operator  s = 1/2 Pauli matrix
        // The ECPso_cart and ECPso_sph integral code evaluates the integrals
        // <i|H^{SO}|j> as shown in NWChem ECP doc
        //   http://www.nwchem-sw.org/index.php/ECP
        // Note when calling ECPso_spinor function to evaluate the integrals in
        // spinor basis pyscf evaluates the expression like
        //      einsum('sxy,spq,xpi,yqj->ij', pauli_matrix, so_sph, ui.conj(), uj)
        // A factor of 1/2 needs to be multiplied for <H^{SO}> integrals in spinor basis.
        // See also the discussion in https://github.com/pyscf/pyscf/issues/378
        //common_fac *= .5; // 1/2 was applied in old implementation.

        const int lilj1 = li + lj + 1;
        int atm_id, lc, lab, lilc1, ljlc1, dlc, im, mq, jfmq;
        int i, j, n, ic, jc;
        int d2, d3;

        double *rur, *radi, *radj, *angi, *angj, *jmm_angj, *rad_all, *buf, *plast;
        MALLOC_INSTACK(rad_all, nci*ncj*(li+lj+1)*(li+ECP_LMAX+1)*(lj+ECP_LMAX+1));
        MARK_STACK;
        MALLOC_INSTACK(angi, (li+1)*nfi*(ECP_LMAX*2+1)*(li+ECP_LMAX+1));
        MALLOC_INSTACK(angj, (lj+1)*nfj*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));
        MALLOC_INSTACK(buf, nfi*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));
        MALLOC_INSTACK(jmm_angj, (lj+1)*nfj*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1)*3);
        MALLOC_INSTACK(buf, nfi*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));

        RESTORE_STACK;
        MALLOC_INSTACK(rur, nrs*(li+lj+1));
        MALLOC_INSTACK(radi, nci*(li+ECP_LMAX+1)*nrs);
        MALLOC_INSTACK(radj, ncj*(lj+ECP_LMAX+1)*nrs);
        lilc1 = li + ECP_LMAX + 1;
        ljlc1 = lj + ECP_LMAX + 1;
        d2 = lilc1 * ljlc1;
        MALLOC_INSTACK(plast, d2);
        char *converged;
        MALLOC_INSTACK(converged, nci*ncj*lilj1);

        double rca[3];
        double rcb[3];
        double dca, dcb, s;
        double *rc, *pradi, *pradj, *prur;
        double *gctrx = gctr;
        double *gctry = gctrx + ngctr;
        double *gctrz = gctry + ngctr;

        int has_value = 0;
        int ecp_lmax[natm];
        for (i = 0; i < natm; i++) {
                ecp_lmax[i] = 0;
        }
        for (i = 0; i < necpbas; i++) {
                n = ecpbas[ATOM_OF +i*BAS_SLOTS];
                ecp_lmax[n] = MAX(ecp_lmax[n], ecpbas[ANG_OF +i*BAS_SLOTS]);
        }

        for (i = 0; i < ngctr*4; i++) { gctr[i] = 0; }

        for (iloc = 0; iloc < nslots; iloc++) {
                if (ecpbas[SO_TYPE_OF+ecploc[iloc]*BAS_SLOTS] != 1) {
                        continue;
                }
                lc = ecpbas[ANG_OF+ecploc[iloc]*BAS_SLOTS];
                if (lc == -1) { // Treat Ul term as L_max in SO-ECP
                        n = ecpbas[ATOM_OF+ecploc[iloc]*BAS_SLOTS];
                        lc = ecp_lmax[n] + 1;
                }
                atm_id = ecpbas[ATOM_OF+ecploc[iloc]*BAS_SLOTS];
                rc = env + atm[PTR_COORD+atm_id*ATM_SLOTS];
                ecpshls = ecploc + iloc;

        if (!check_3c_overlap(shls, atm, bas, env, rc, ecpshls, ecpbas)) {
                continue;
        }

        has_value = 1;
        rca[0] = rc[0] - ri[0];
        rca[1] = rc[1] - ri[1];
        rca[2] = rc[2] - ri[2];
        rcb[0] = rc[0] - rj[0];
        rcb[1] = rc[1] - rj[1];
        rcb[2] = rc[2] - rj[2];
        dca = sqrt(SQUARE(rca));
        dcb = sqrt(SQUARE(rcb));

        dlc = lc * 2 + 1;
        lilc1 = li + lc + 1;
        ljlc1 = lj + lc + 1;
        d2 = lilc1 * ljlc1;
        d3 = lilj1 * d2;
        im = nfi * dlc;
        mq = dlc * ljlc1;
        jfmq = (lj+1) * nfj * mq;

        int level, nrs0, start, step, ijl;
        double wtscale;
        double *prad;
        char all_conv;
        double *rs = rs_gauss_chebyshev2047;
        double *ws = ws_gauss_chebyshev2047;
        for (i = 0; i < nci*ncj*lilj1; i++) { converged[i] = 0; }
        for (i = 0; i < nci*ncj*d3; i++) { rad_all[i] = 0; }
        nrs0 = (1 << LEVEL0) - 1;
        step = 1 << (LEVEL_MAX - LEVEL0);
        start = step - 1;
        wtscale = step;
        for (level = LEVEL0; level <= LEVEL_MAX; level++) {
                nrs = ECPrad_part(rur, rs, start, nrs0, step, ecpshls, ecpbas,
                                  atm, natm, bas, nbas, env, opt, cache);
                for (i = 0; i < nrs; i++) {
                        rur[i] *= ws[start+i*step] * wtscale;
                        for (lab = 1; lab <= li+lj; lab++) {
                                rur[nrs*lab+i] = rur[nrs*(lab-1)+i] * rs[start+i*step];
                        }
                }

                type2_facs_rad(radi, ish, lc, dca, rs+start, nrs, step,
                               atm, natm, bas, nbas, env, cache);
                type2_facs_rad(radj, jsh, lc, dcb, rs+start, nrs, step,
                               atm, natm, bas, nbas, env, cache);
                all_conv = 1;
                for (ijl = 0, ic = 0; ic < nci; ic++) {
                for (jc = 0; jc < ncj; jc++) {
                        pradi = radi + ic * nrs * lilc1;
                        pradj = radj + jc * nrs * ljlc1;
                        for (lab = 0; lab <= li+lj; lab++, ijl++) {
                                if (!converged[ijl]) {
        prur = rur + lab * nrs;
        prad = rad_all + ijl*d2;
        for (i = 0; i < d2; i++) {
                plast[i] = prad[i];
                prad[i] *= .5;
        }

        for (i = 0; i < lilc1; i++) {
        for (j = 0; j < ljlc1; j++) {
                s = prad[i*ljlc1+j];
                for (n = 0; n < nrs; n++) {
                        s += prur[n] * pradi[n*lilc1+i] * pradj[n*ljlc1+j];
                }
                prad[i*ljlc1+j] = s;
        } }

        for (i = 0; i < d2; i++) {
                if (!CLOSE_ENOUGH(plast[i], prad[i])) {
                        converged[ijl] = 0;
                        all_conv = 0;
                        break;
                }
        }
                                }
                        }
                } }

                if (all_conv) {
                        break;
                } else {
                        nrs0 = (1 << level) - 1;
                        step = 1 << (LEVEL_MAX - level);
                        start = (start - 1) / 2;
                        wtscale *= .5;
                }
        }

        type2_facs_ang(angi, li, lc, rca, cache);
        type2_facs_ang(angj, lj, lc, rcb, cache);
        transform_angj(jmm_angj, angj, lj, lc);

        for (ic = 0; ic < nci; ic++) {
        for (jc = 0; jc < ncj; jc++) {
                prad = rad_all + (ic*ncj+jc)*d3;
                for (i = 0; i <= li; i++) {
                for (j = 0; j <= lj; j++) {
                        dgemm_(&TRANS_N, &TRANS_N, &ljlc1, &im, &lilc1,
                               &D1, prad+(i+j)*d2, &ljlc1,
                               angi+i*nfi*dlc*lilc1, &lilc1, &D0, buf, &ljlc1);
                        dgemm_(&TRANS_T, &TRANS_N, &nfi, &nfj, &mq,
                               &common_fac, buf, &mq, jmm_angj       +j*nfj*dlc*ljlc1, &mq,
                               &D1, gctrx+jc*nfj*di+ic*nfi, &di);
                        dgemm_(&TRANS_T, &TRANS_N, &nfi, &nfj, &mq,
                               &common_fac, buf, &mq, jmm_angj+jfmq  +j*nfj*dlc*ljlc1, &mq,
                               &D1, gctry+jc*nfj*di+ic*nfi, &di);
                        dgemm_(&TRANS_T, &TRANS_N, &nfi, &nfj, &mq,
                               &common_fac, buf, &mq, jmm_angj+jfmq*2+j*nfj*dlc*ljlc1, &mq,
                               &D1, gctrz+jc*nfj*di+ic*nfi, &di);
                } }
        } }
        }
        return has_value;
}

static void scale_coeff(double *cei, const double *ci, const double *ai,
                        const double r2ca, const int npi, const int nci, const int li)
{
        int ip, ic;
        double tmp;
        double common_fac = CINTcommon_fac_sp(li) * 4 * M_PI;
        for (ip = 0; ip < npi; ip++) {
                tmp = exp(-ai[ip] * r2ca) * common_fac;
                for (ic = 0; ic < nci; ic++) {
                        cei[ic*npi+ip] = ci[ic*npi+ip] * tmp;
                }
        }
}

void type1_rad_part(double *rad_all, int lmax, double k, double aij,
                    double *ur, double *rs, int nrs, int inc, double *cache)
{
        if (nrs == 0) {
                return;
        }

        const int lmax1 = lmax + 1;
        double *rur, *bval;
        MALLOC_INSTACK(rur, nrs);
        MALLOC_INSTACK(bval, nrs*lmax1);
        int lab, i, n;
        double kaij, fac, tmp, s;
        double *prad;

        kaij = k / (2*aij);
        fac = kaij*kaij*aij;
        for (n = 0; n < nrs; n++) {
                tmp = rs[n*inc] - kaij;
                tmp = fac - aij*tmp*tmp;
                if (ur[n] == 0
/* Avoid exp(tmp) to be infinity!  Usually such points correspond to remote
 * functions.  Most likely, they can be dropped without affecting accuracy. */
                    || tmp > CUTOFF
/* + 6. to include the largest value in ECPsph_ine_opt
 * +30. to include the rs^lmax of remote point for high-l functions. */
                    || tmp < -(EXPCUTOFF+6.+30.)) {
                        rur[n] = 0;
                        for (i = 0; i < lmax1; i++) {
                                bval[n*lmax1+i] = 0;
                        }
                } else {
                        rur[n] = ur[n] * exp(tmp);
                        ECPsph_ine_opt(bval+n*lmax1, lmax, k*rs[n*inc]);
                }
        }

        for (lab = 0; lab <= lmax; lab++) {
                if (lab > 0) {
                        for (n = 0; n < nrs; n++) {
                                rur[n] *= rs[n*inc];
                        }
                }
                prad = rad_all + lab * lmax1;
                for (i = lab%2; i <= lmax; i+=2) {
                        s = prad[i];
                        for (n = 0; n < nrs; n++) {
                                s += rur[n] * bval[n*lmax1+i];
                        }
                        prad[i] = s;
                }
        }
}

int ECPtype1_cart(double *gctr, int *shls, int *ecpbas, int necpbas,
                  int *atm, int natm, int *bas, int nbas, double *env,
                  ECPOpt *opt, double *cache)
{
        if (necpbas == 0) {
                return 0;
        }
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int npi = bas[NPRIM_OF+ish*BAS_SLOTS];
        const int npj = bas[NPRIM_OF+jsh*BAS_SLOTS];
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int ngctr = nci * ncj * nfi * nfj;
        const double *ai = env + bas[PTR_EXP+ish*BAS_SLOTS];
        const double *aj = env + bas[PTR_EXP+jsh*BAS_SLOTS];
        const double *ci = env + bas[PTR_COEFF+ish*BAS_SLOTS];
        const double *cj = env + bas[PTR_COEFF+jsh*BAS_SLOTS];
        const double *ri = env + atm[PTR_COORD+bas[ATOM_OF+ish*BAS_SLOTS]*ATM_SLOTS];
        const double *rj = env + atm[PTR_COORD+bas[ATOM_OF+jsh*BAS_SLOTS]*ATM_SLOTS];

        int nrs = 1 << LEVEL_MAX;
        int *ecploc;
        MALLOC_INSTACK(ecploc, necpbas+1);
        int nslots = _loc_ecpbas(ecploc, ecpbas, necpbas);
        int *ecpshls;
        int iloc;

        const int lilj1 = li + lj + 1;
        const int d1 = lilj1;
        const int d2 = d1 * d1;
        const int d3 = d2 * d1;
        const int di1 = li + 1;
        const int di2 = di1 * di1;
        const int di3 = di2 * di1;
        const int dj1 = lj + 1;
        const int dj2 = dj1 * dj1;
        const int dj3 = dj2 * dj1;
        int atm_id;
        int i, n, ip, jp, ic, jc, mi, mj;
        int i1, i2, i3, j1, j2, j3;
        int ix, iy, iz, jx, jy, jz;
        double *ur, *rad_all, *rad_ang, *rad_ang_all, *ifac, *jfac, *cei, *cej, *plast;
        MALLOC_INSTACK(ur, nrs);
        MALLOC_INSTACK(rad_all, npi*npj*d2);
        MALLOC_INSTACK(rad_ang, d3);
        MALLOC_INSTACK(rad_ang_all, nci*ncj*d3);
        MALLOC_INSTACK(ifac, nfi*di3);
        MALLOC_INSTACK(jfac, nfj*dj3);
        MALLOC_INSTACK(cei, npi*nci);
        MALLOC_INSTACK(cej, npj*ncj);
        MALLOC_INSTACK(plast, d2);
        double rca[3];
        double rcb[3];
        double rij[3];
        double fac;
        double *rc, *pifac, *pjfac, *pout;
        int has_value = 0;
        char *converged;
        MALLOC_INSTACK(converged, npi*npj);

        for (i = 0; i < ngctr; i++) { gctr[i] = 0; }
        for (i = 0; i < d3; i++) { rad_all[i] = 0; }

        for (iloc = 0; iloc < nslots; iloc++) {
                if (ecpbas[ANG_OF+ecploc[iloc]*BAS_SLOTS] != -1) {
                        continue;
                }
                atm_id = ecpbas[ATOM_OF+ecploc[iloc]*BAS_SLOTS];
                rc = env + atm[PTR_COORD+atm_id*ATM_SLOTS];
                ecpshls = ecploc + iloc;

        if (!check_3c_overlap(shls, atm, bas, env, rc, ecpshls, ecpbas)) {
                continue;
        }

        has_value = 1;
        rca[0] = rc[0] - ri[0];
        rca[1] = rc[1] - ri[1];
        rca[2] = rc[2] - ri[2];
        rcb[0] = rc[0] - rj[0];
        rcb[1] = rc[1] - rj[1];
        rcb[2] = rc[2] - rj[2];
        scale_coeff(cei, ci, ai, SQUARE(rca), npi, nci, li);
        scale_coeff(cej, cj, aj, SQUARE(rcb), npj, ncj, lj);

        int level, nrs0, start, step;
        double wtscale;
        double *prad;
        char all_conv;
        double *rs = rs_gauss_chebyshev2047;
        double *ws = ws_gauss_chebyshev2047;
        for (i = 0; i < npi*npj; i++) { converged[i] = 0; }
        for (i = 0; i < npi*npj*d2; i++) { rad_all[i] = 0; }
        nrs0 = (1 << LEVEL0) - 1;
        step = 1 << (LEVEL_MAX - LEVEL0);
        start = step - 1;
        wtscale = step;
        for (level = LEVEL0; level <= LEVEL_MAX; level++) {
                nrs = ECPrad_part(ur, rs, start, nrs0, step, ecpshls, ecpbas,
                                  atm, natm, bas, nbas, env, opt, cache);
                if (nrs == 0) {
                        break;
                }
                for (n = 0; n < nrs; n++) {
                        ur[n] *= ws[start+n*step] * wtscale;
                }

                all_conv = 1;
                for (ip = 0; ip < npi; ip++) {
                for (jp = 0; jp < npj; jp++) {
                        if (!converged[ip*npj+jp]) {
                                prad = rad_all + (ip*npj+jp)*d2;
                                for (i = 0; i < d2; i++) {
                                        plast[i] = prad[i];
                                        prad[i] *= .5;
                                }
                                rij[0] = ai[ip] * rca[0] + aj[jp] * rcb[0];
                                rij[1] = ai[ip] * rca[1] + aj[jp] * rcb[1];
                                rij[2] = ai[ip] * rca[2] + aj[jp] * rcb[2];
                                type1_rad_part(prad, li+lj, sqrt(SQUARE(rij))*2,
                                               ai[ip]+aj[jp], ur, rs+start, nrs, step, cache);
                                converged[ip*npj+jp] = 1;
                                for (i = 0; i < d2; i++) {
                                        if (!CLOSE_ENOUGH(plast[i],prad[i])) {
                                                converged[ip*npj+jp] = 0;
                                                all_conv = 0;
                                                break;
                                        }
                                }
                        }
                } }

                if (all_conv) {
                        break;
                } else {
                        nrs0 = (1 << level) - 1;
                        step = 1 << (LEVEL_MAX - level);
                        start = (start - 1) / 2;
                        wtscale *= .5;
                }
        }

        for (i = 0; i < nci*ncj*d3; i++) { rad_ang_all[i] = 0; }
        for (ip = 0; ip < npi; ip++) {
        for (jp = 0; jp < npj; jp++) {
                rij[0] = ai[ip] * rca[0] + aj[jp] * rcb[0];
                rij[1] = ai[ip] * rca[1] + aj[jp] * rcb[1];
                rij[2] = ai[ip] * rca[2] + aj[jp] * rcb[2];
                type1_rad_ang(rad_ang, li+lj, rij, rad_all+(ip*npj+jp)*d2);
                for (ic = 0; ic < nci; ic++) {
                for (jc = 0; jc < ncj; jc++) {
                        fac = cei[ic*npi+ip] * cej[jc*npj+jp];
                        prad = rad_ang_all + (ic*ncj+jc)*d3;
                        for (n = 0; n < d3; n++) {
                                prad[n] += fac * rad_ang[n];
                        }
                } }
        } }

        type1_static_facs(ifac, li, rca, cache);
        type1_static_facs(jfac, lj, rcb, cache);
        for (ic = 0; ic < nci; ic++) {
        for (jc = 0; jc < ncj; jc++) {
                prad = rad_ang_all + (ic*ncj+jc)*d3;
                LOOP_CART(li, mi, ix, iy, iz) {
                LOOP_CART(lj, mj, jx, jy, jz) {
                        pifac = ifac + mi * di3;
                        pjfac = jfac + mj * dj3;
                        pout = gctr + (jc*nfj+mj) * nci*nfi + ic*nfi+mi;
                        LOOP_XYZ(i1, i2, i3, ix, iy, iz) {
                        LOOP_XYZ(j1, j2, j3, jx, jy, jz) {
                                *pout += pifac[i1*di2+i2*di1+i3] *
                                         pjfac[j1*dj2+j2*dj1+j3] *
                                         prad[(i1+j1)*d2+(i2+j2)*d1+i3+j3];
                        } }
                } }
        } }

        }
        return has_value;
}

int ECPscalar_cache_size(int comp, int *shls,
                         int *atm, int natm, int *bas, int nbas, double *env)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int npi = bas[NPRIM_OF+ish*BAS_SLOTS];
        const int npj = bas[NPRIM_OF+jsh*BAS_SLOTS];
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int lilj1 = li + lj + 1;
        const int lilc1 = li + ECP_LMAX + 1;
        const int ljlc1 = lj + ECP_LMAX + 1;
        const int d1 = lilj1;
        const int d2 = d1 * d1;
        const int d3 = d2 * d1;
        const int di1 = li + 1;
        const int di2 = di1 * di1;
        const int di3 = di2 * di1;
        const int dj1 = lj + 1;
        const int dj2 = dj1 * dj1;
        const int dj3 = dj2 * dj1;
        const int nrs =  1 << LEVEL_MAX;
        int size1 = nci*ncj*(li+lj+1);
        size1 += ((li+1)*nfi*(ECP_LMAX*2+1)*(li+ECP_LMAX+1) +
                  (lj+1)*nfj*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1));
        size1 += nfi*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1);
        size1 += (lj+1)*nfj*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1)*3;
        int size2 = nrs * (li+lj+1 + 1 +
                           nci*(li+ECP_LMAX+1) +
                           ncj*(lj+ECP_LMAX+1) +
                           MAX(npi*lilc1, npj*ljlc1));
        size2 += lilc1 * ljlc1;
        size2 += MAX(di1*di1*di1*lilc1, dj1*dj1*dj1*ljlc1) * (ECP_LMAX*2 + 1);
        int size = nfi*nfj*(nci*ncj+2) * comp;
        size += nci*ncj*(li+lj+1)*(li+ECP_LMAX+1)*(lj+ECP_LMAX+1);
        size += MAX(size1, size2);
        size += nfi*(ECP_LMAX*2+1)*(lj+ECP_LMAX+1);
        size += npi*npj*d2;
        size += d3;
        size += nci*ncj*d3;
        size += nfi*di3;
        size += nfj*dj3;
        size += npi*nci;
        size += npj*ncj;
        size += natm;
        return size;
}

int ECPscalar_c2s_factory(Function_cart fcart, double *gctr, int comp, int *shls,
                          int *ecpbas, int necpbas, int *atm, int natm,
                          int *bas, int nbas, double *env, ECPOpt *opt,
                          double *cache)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];

        if (li < 2 && lj < 2) {
                return fcart(gctr, shls, ecpbas, necpbas,
                             atm, natm, bas, nbas, env, opt, cache);
        }

        int j;
        int di = nfi * nci;
        int dji = di * (lj*2+1);
        int nij = (li*2+1) * (lj*2+1) * nci * ncj;
        double *gcart, *gtmp;
        MALLOC_INSTACK(gcart, nfi*nfj*nci*ncj * comp);
        gtmp = cache;
        int has_value = fcart(gcart, shls, ecpbas, necpbas,
                              atm, natm, bas, nbas, env, opt, cache);

        if (has_value) {
                if (li < 2) {
                        for (j = 0; j < ncj * comp; j++) {
                                CINTc2s_ket_sph(gctr+j*dji, di, gcart+j*nfj*di, lj);
                        }
                } else if (lj < 2 ) {
                        CINTc2s_bra_sph(gctr, (lj*2+1)*nci*ncj * comp, gcart, li);
                } else {
                        for (j = 0; j < ncj * comp; j++) {
                                CINTc2s_ket_sph(gtmp+j*dji, di, gcart+j*nfj*di, lj);
                        }
                        CINTc2s_bra_sph(gctr, (lj*2+1)*nci*ncj * comp, gtmp, li);
                }
        } else {
                for (j = 0; j < nij * comp; j++) { gctr[j] = 0; }
        }
        return has_value;
}

int ECPtype1_sph(double *gctr, int *shls, int *ecpbas, int necpbas,
                 int *atm, int natm, int *bas, int nbas, double *env,
                 ECPOpt *opt, double *cache)
{
        return ECPscalar_c2s_factory(ECPtype1_cart, gctr, 1, shls, ecpbas, necpbas,
                                     atm, natm, bas, nbas, env, opt, cache);
}

int ECPtype2_sph(double *gctr, int *shls, int *ecpbas, int necpbas,
                 int *atm, int natm, int *bas, int nbas, double *env,
                 ECPOpt *opt, double *cache)
{
        return ECPscalar_c2s_factory(ECPtype2_cart, gctr, 1, shls, ecpbas, necpbas,
                                     atm, natm, bas, nbas, env, opt, cache);
}

void ECPscalar_distribute(double *out, double *gctr, const int *dims,
                          const int comp, const int di, const int dj)
{
        const int dij = di * dj;
        int i, j, ni, icomp;
        if (dims == NULL) {
                for (i = 0; i < dij * comp; i++) {
                        out[i] = gctr[i];
                }
        } else {
                ni = dims[0];
                for (icomp = 0; icomp < comp; icomp++) {
                        for (i = 0; i < di; i++) {
                        for (j = 0; j < dj; j++) {
                                out[i+j*ni] = gctr[i+j*di];
                        } }
                        out += dims[0] * dims[1];
                        gctr += dij;
                }
        }
}
void ECPscalar_distribute0(double *out, const int *dims,
                           const int comp, const int di, const int dj)
{
        const int dij = di * dj;
        int i, j, ni, icomp;
        if (dims == NULL) {
                for (i = 0; i < dij * comp; i++) {
                        out[i] = 0;
                }
        } else {
                ni = dims[0];
                for (icomp = 0; icomp < comp; icomp++) {
                        for (i = 0; i < di; i++) {
                        for (j = 0; j < dj; j++) {
                                out[i+j*ni] = 0;
                        } }
                        out += dims[0] * dims[1];
                }
        }
}


/*
 * For moleintor.getints function
 */
int ECPscalar_sph(double *out, int *dims, int *shls, int *atm, int natm,
                  int *bas, int nbas, double *env, ECPOpt *opt, double *cache)

{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int di = (li*2+1) * bas[NCTR_OF+ish*BAS_SLOTS];
        const int dj = (lj*2+1) * bas[NCTR_OF+jsh*BAS_SLOTS];
        const int dij = di * dj;
        const int comp = 1;

        if (out == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                return cache_size;
        }
        double *stack = NULL;
        if (cache == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                stack = malloc(sizeof(double) * cache_size);
                cache = stack;
        }

        int *ecpbas = bas + (int)(env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)(env[AS_NECPBAS]);
        double *buf1, *buf2;
        MALLOC_INSTACK(buf1, dij * comp);
        MALLOC_INSTACK(buf2, dij * comp);
        int has_value;
        has_value = ECPtype1_sph(buf1, shls, ecpbas, necpbas,
                                 atm, natm, bas, nbas, env, opt, cache);
        has_value =(ECPtype2_sph(buf2, shls, ecpbas, necpbas,
                                 atm, natm, bas, nbas, env, opt, cache) | has_value);

        if (has_value) {
                int i;
                for (i = 0; i < dij; i++) {
                        buf1[i] += buf2[i];
                }
                ECPscalar_distribute(out, buf1, dims, 1, di, dj);
        } else {
                ECPscalar_distribute0(out, dims, 1, di, dj);
        }

        if (stack != NULL) {
                free(stack);
        }
        return has_value;
}

int ECPscalar_cart(double *out, int *dims, int *shls, int *atm, int natm,
                   int *bas, int nbas, double *env, ECPOpt *opt, double *cache)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int di = (li+1) * (li+2) / 2 * bas[NCTR_OF+ish*BAS_SLOTS];
        const int dj = (lj+1) * (lj+2) / 2 * bas[NCTR_OF+jsh*BAS_SLOTS];
        const int dij = di * dj;
        const int comp = 1;

        if (out == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                return cache_size;
        }
        double *stack = NULL;
        if (cache == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                stack = malloc(sizeof(double) * cache_size);
                cache = stack;
        }

        int *ecpbas = bas + ((int)env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)env[AS_NECPBAS];
        double *buf1, *buf2;
        MALLOC_INSTACK(buf1, dij * comp);
        MALLOC_INSTACK(buf2, dij * comp);
        int has_value;
        has_value = ECPtype1_cart(buf1, shls, ecpbas, necpbas,
                                  atm, natm, bas, nbas, env, opt, cache);
        has_value =(ECPtype2_cart(buf2, shls, ecpbas, necpbas,
                                  atm, natm, bas, nbas, env, opt, cache) | has_value);

        if (has_value) {
                int i;
                for (i = 0; i < dij; i++) {
                        buf1[i] += buf2[i];
                }
                ECPscalar_distribute(out, buf1, dims, 1, di, dj);
        } else {
                ECPscalar_distribute0(out, dims, 1, di, dj);
        }

        if (stack != NULL) {
                free(stack);
        }
        return has_value;
}

/* Port from libcint */
void CINTc2s_bra_spinor_si(double complex *gsp, int nket,
                           double complex *gcart, int kappa, int l);
void CINTc2s_ket_spinor(double complex *gsp, int nbra,
                        double complex *gcart, int kappa, int l);
void CINTdcmplx_pp(const int n, double complex *z,
                   const double *re, const double *im);
void CINTdcmplx_pn(const int n, double complex *z,
                   const double *re, const double *im);
void CINTdcmplx_np(const int n, double complex *z,
                   const double *re, const double *im);
static void zcopy_ij(double complex *opij, const double complex *gctr,
                     const int ni, const int nj, const int mi, const int mj)
{
        int i, j;

        for (j = 0; j < mj; j++) {
                for (i = 0; i < mi; i++) {
                        opij[j*ni+i] = gctr[j*mi+i];
                }
        }
}
// multiply pauli_matrix
static void cart2spinor(double complex *opij, double *gctr, int *dims,
                        int *shls, int *bas, double *cache)
{
        //TODO: call libcint function c2s_si_1ei. Note c2s_si_1ei multiplies
        //-1*pauli_matrix than pauli_matrix

        int i_sh = shls[0];
        int j_sh = shls[1];
        int li = bas[ANG_OF+i_sh*BAS_SLOTS];
        int lj = bas[ANG_OF+j_sh*BAS_SLOTS];
        int i_kp = bas[KAPPA_OF+i_sh*BAS_SLOTS];
        int j_kp = bas[KAPPA_OF+j_sh*BAS_SLOTS];
        int i_ctr = bas[NCTR_OF+i_sh*BAS_SLOTS];
        int j_ctr = bas[NCTR_OF+j_sh*BAS_SLOTS];
        int di = CINTlen_spinor(i_sh, bas);
        int dj = CINTlen_spinor(j_sh, bas);
        int ni = dims[0];
        int nj = dims[1];
        int ofj = ni * dj;
        int nfi = (li+1)*(li+2)/2;
        int nfj = (lj+1)*(lj+2)/2;
        int nf2i = nfi + nfi;
        int nf2j = nfj + nfj;
        int nf = nfi * nfj;
        int ic, jc;
        double *gc_x = gctr;
        double *gc_y = gc_x + nf * i_ctr * j_ctr;
        double *gc_z = gc_y + nf * i_ctr * j_ctr;
        double *gc_1 = gc_z + nf * i_ctr * j_ctr;
        double complex *tmp1, *tmp2;
        MALLOC_INSTACK(tmp1, nf2i*nf2j);
        MALLOC_INSTACK(tmp2, nf2i*nf2j);

        for (jc = 0; jc < j_ctr; jc++) {
        for (ic = 0; ic < i_ctr; ic++) {
                //cmplx( gctr.POS_Z, gctr.POS_1)
                //cmplx( gctr.POS_X,-gctr.POS_Y)
                CINTdcmplx_pp(nf, tmp1, gc_z, gc_1);
                CINTdcmplx_pn(nf, tmp1+nf, gc_x, gc_y);
                //cmplx( gctr.POS_X, gctr.POS_Y)
                //cmplx(-gctr.POS_Z, gctr.POS_1)
                CINTdcmplx_pp(nf, tmp1+nfi*nf2j, gc_x, gc_y);
                CINTdcmplx_np(nf, tmp1+nfi*nf2j+nf, gc_z, gc_1);
                CINTc2s_bra_spinor_si(tmp2, nf2j, tmp1, i_kp, li);
                CINTc2s_ket_spinor(tmp1, di, tmp2, j_kp, lj);
                zcopy_ij(opij+ofj*jc+di*ic, tmp1, ni, nj, di, dj);

                gc_x += nf;
                gc_y += nf;
                gc_z += nf;
                gc_1 += nf;
        } }
}

void ECPscalar_optimizer(ECPOpt **opt, int *atm, int natm, int *bas, int nbas, double *env)
{
        ECPOpt *opt0 = (ECPOpt *)malloc(sizeof(ECPOpt));
        *opt = opt0;

        int *ecpbas = bas + (int)(env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)(env[AS_NECPBAS]);

        double r2, s;
        double *ak, *ck, *uk;
        int npk;
        int i, ib, kp;

        opt0->u_ecp = malloc(sizeof(double) * (1 << LEVEL_MAX) * necpbas);
        uk = opt0->u_ecp;
        for (ib = 0; ib < necpbas; ib++) {
                npk = ecpbas[ib*BAS_SLOTS+NPRIM_OF];
                ak = env + ecpbas[ib*BAS_SLOTS+PTR_EXP];
                ck = env + ecpbas[ib*BAS_SLOTS+PTR_COEFF];

                for (i = 0; i < (1 << LEVEL_MAX); i++) {
                        r2 = rs_gauss_chebyshev2047[i]*rs_gauss_chebyshev2047[i];
                        uk[i] = ck[0] * exp(-ak[0]*r2);
                        s = uk[i];
                        for (kp = 1; kp < npk; kp++) {
                                s += ck[kp] * exp(-ak[kp]*r2);
                        }
                        uk[i] = s;
                }
                uk += (1 << LEVEL_MAX);
        }
}

void ECPdel_optimizer(ECPOpt **opt)
{
        if (opt == NULL) {
                return;
        }

        ECPOpt *opt0 = *opt;
        if (opt0 == NULL) {
                return;
        }

        if (opt0->u_ecp != NULL) {
                free(opt0->u_ecp);
        }
        free(opt0);
        opt = NULL;
}


/*
 * <i| l U(r)|j>
 * H^{SO} integrals in spinor basis can be evaluated
 *      .5 * einsum('sxy,spq,xpi,yqj->ij', pauli_matrix, so_cart, ui.conj(), uj)
 */
int ECPso_cart(double *out, int *dims, int *shls, int *atm, int natm,
               int *bas, int nbas, double *env, ECPOpt *opt, double *cache)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int di = (li+1) * (li+2) / 2 * bas[NCTR_OF+ish*BAS_SLOTS];
        const int dj = (lj+1) * (lj+2) / 2 * bas[NCTR_OF+jsh*BAS_SLOTS];
        const int dij = di * dj;
        const int comp = 4;

        if (out == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                return cache_size;
        }
        double *stack = NULL;
        if (cache == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                stack = malloc(sizeof(double) * cache_size);
                cache = stack;
        }

        int *ecpbas = bas + ((int)env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)env[AS_NECPBAS];
        double *buf1;
        MALLOC_INSTACK(buf1, dij * comp);
        int has_value = ECPtype_so_cart(buf1, shls, ecpbas, necpbas,
                                        atm, natm, bas, nbas, env, opt, cache);

        if (has_value) {
                // comp-1 for x, y, z components only
                ECPscalar_distribute(out, buf1, dims, comp-1, di, dj);
        } else {
                ECPscalar_distribute0(out, dims, comp-1, di, dj);
        }

        if (stack != NULL) {
                free(stack);
        }
        return has_value;
}

int ECPso_sph(double *out, int *dims, int *shls, int *atm, int natm,
              int *bas, int nbas, double *env, ECPOpt *opt, double *cache)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int di = (li*2+1) * nci;
        const int dj = (lj*2+1) * ncj;
        const int dij = di * dj;
        const int comp = 4;

        if (out == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                return cache_size;
        }
        double *stack = NULL;
        if (cache == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                stack = malloc(sizeof(double) * cache_size);
                cache = stack;
        }

        int *ecpbas = bas + (int)(env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)(env[AS_NECPBAS]);
        double *buf1;
        MALLOC_INSTACK(buf1, dij * comp);
        int has_value = ECPscalar_c2s_factory(ECPtype_so_cart, buf1, comp,
                                              shls, ecpbas, necpbas,
                                              atm, natm, bas, nbas, env, opt, cache);

        if (has_value) {
                // comp-1 for x, y, z components only
                ECPscalar_distribute(out, buf1, dims, comp-1, di, dj);
        } else {
                ECPscalar_distribute0(out, dims, comp-1, di, dj);
        }

        if (stack != NULL) {
                free(stack);
        }
        return has_value;
}

#define OF_CMPLX 2
int ECPso_spinor(double complex *out, int *dims, int *shls, int *atm, int natm,
                 int *bas, int nbas, double *env, ECPOpt *opt, double *cache)
{
        const int ish = shls[0];
        const int jsh = shls[1];
        const int li = bas[ANG_OF+ish*BAS_SLOTS];
        const int lj = bas[ANG_OF+jsh*BAS_SLOTS];
        const int nfi = (li+1) * (li+2) / 2;
        const int nfj = (lj+1) * (lj+2) / 2;
        const int nci = bas[NCTR_OF+ish*BAS_SLOTS];
        const int ncj = bas[NCTR_OF+jsh*BAS_SLOTS];
        const int ngctr = nfi * nfj * nci * ncj;
        const int di = CINTcgto_spinor(shls[0], bas);
        const int dj = CINTcgto_spinor(shls[1], bas);
        const int comp = 4;

        if (out == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                return cache_size + ngctr*8*OF_CMPLX;
        }
        double *stack = NULL;
        if (cache == NULL) {
                int cache_size = ECPscalar_cache_size(comp, shls,
                                                      atm, natm, bas, nbas, env);
                stack = malloc(sizeof(double) * (cache_size + ngctr*8*OF_CMPLX));
                cache = stack;
        }

        int *ecpbas = bas + (int)(env[AS_ECPBAS_OFFSET])*BAS_SLOTS;
        int necpbas = (int)(env[AS_NECPBAS]);
        double *buf1;
        MALLOC_INSTACK(buf1, ngctr * comp);
        int has_value = ECPtype_so_cart(buf1, shls, ecpbas, necpbas,
                                        atm, natm, bas, nbas, env, opt, cache);
        int counts[2] = {di, dj};
        if (dims == NULL) {
                dims = counts;
        }
        cart2spinor(out, buf1, dims, shls, bas, cache);

        if (stack != NULL) {
                free(stack);
        }
        return has_value;
}
